Benchmark three-dimensional elasticity solutions for the buckling of thick orthotropic cylindrical shells

Benchmark solutions to the problem of buckling of orthotropic cylindrical shells, which are based on the three-dimensional theory of elasticity, are presented in this review article. It is assumed that the shell is under external pressure or axial compression or a combination of these loadings. These solutions provide a means of accurately assessing the limitations of the various shell theories in predicting critical loads. A comparison with some classical shell theories shows that the classical shell theories may produce, in general, highly non-conservative results on the critical load of composite shells with thick construction. One noteworthy exception: the Timoshenko shell buckling equations produce conservative results under pure axial compression.     

Exact characteristic equations for free vibrations of thin orthotropic circular cylindrical shells

This paper presents an analytical procedure and closed-form vibration solutions with analytically determined coefficients for orthotropic circular cylindrical shells having classical boundary conditions. This analysis is based upon the Donnell-Mushtari shell theory. This is the simplest thin shell theory and its results for the lowest frequencies of a closed cylinder may not be as accurate. It is known that the exact procedure is complicated for orthotropic shells and this complexity has apparently prevented most researchers from getting results. Using the separation of variables method, the closed-form natural frequencies are successfully obtained in this work. They are found in a compact form. Moreover, the characteristics of the eigenvalues are examined. The exact solutions are validated through numerical comparisons with available solutions in literatures and the semi-analytical differential quadrature finite element method (S-DQFEM) solutions calculated by the authors.     

Buckling of thick orthotropic spherical shells

This paper is concerned with the semi-analytical solution for the axisymmetric buckling for perfect complete thick orthotropic spherical shells and hemispherical shells under various edge conditions, subjected to uniform full external pressures. The meridional and radial mid-surface displacements and the circumferential cross-sectional rotation are expressed as an expansion in Legendre polynomials respectively in the meridional coordinate. Transverse shear strain effects are included in the energy formulation of small strains. The solutions are achieved directly by use of the Ritz method without considering the force and moment equilibrium and solving the complicated governing equations. Critical buckling loads and the various modes are found from the equations by use of orthogonality and some integral relations.     

粘弹性复合材料层合板壳的动力稳定性分析

分析面内周期激励下粘弹性层合平板以及轴向周期荷载作用下粘弹性层合圆柱壳的动力稳定性。设粘弹性复合材料服从Boltzmann积分型本构关系，其松弛模量由Prony—Dirichlet级数表示，基于薄板与薄壳理论，分别得到对称正交铺设层合板与层合圆柱壳的微分-积分型动力学方程，并应用谐波平衡法直接求解，忽略积分运算所产生的衰减项，导出确定动力不稳定区域边界的特征方程。分析结果表明，主要动力不稳定区域的缩小与材料的粘性参数以及结构横向振动的基频密切相关。     

Dynamic instability of suddenly heated, thick, composite shells

The parametric resonance type of dynamic instability is investigated for suddenly heated, long circular cylindrical shells of composite material. Particular attention is devoted to the effects of thickness shear flexibility and to mid-surface extensibility. Both long-term and short-term responses are investigated.     

Higher order theories for thick cylindrical shells

Summary A variational derivation is used to obtain 10th and 12th order shell theories along with the associated boundary conditions. A computer program produces the coefficients in the reductions of the sets of equilibrium equations to equivalent single 10th and 12th order equations. Exact solutions for closed shells which decay in the axial direction are obtained and compared with exact three dimensional solutions in order to assess the accuracy of the shell theory as an approximation to the elasticity theory.     

Buckling of composite orthotropic cylindrical shells under non-uniform axial loads

The static buckling of orthotropic composite cylindrical shells, under circumferentially non-uniform axial loads is investigated based on Flügge-type field equations. Use of a complex finite Fourier transform provides a simple method for handling any arbitrary non-uniform load but introduces modal coupling between the transformed equations. For simply supported boundaries (conditions SS3) the determination of the critical buckling load reduces to finding the eigenvalues of a finite matrix. Three different non-uniform loads are considered, having forms proportional to (1 + 2cosθ), cosθ and (θ^* − θ) where is the Heaviside function, θ is the circumferential coordinate and aθ^* is the width of an axially loaded strip of the shell of radius a. Computed results indicate the sensitivity of the critical buckling loads to the type of non-uniform load and the material lay-ups of the cylinders.     

Vibrations of axially compressed laminated orthotropic cylindrical shells,including transverse shear deformation

Summary The free vibrations, buckling and the effect of initial prestress upon the frequency spectrum of orthotropic composite cylindrical shells are examined in the context of a theory that includes transverse shear deformation. Results obtained are compared with the predictions of refined Love-type theory and simplified Donnell-type theory that do not consider shear deformation.The calculated examples indicate that transverse shear deformation can be significant not only for short composite shells but even for longer shells possessing low shear moduli.

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Schwingungen axial gedrückter, laminierter, orthotroper zylindrischer Schalen einschließlich Querschubverformung

Zusammenfassung Es werden die freien Schwingungen, das Beulen und der Einfluß einer Anfangsvorspannung auf das Frequenzspektrum orthotroper, kompositer, zylindrischer Schalen im Rahmen einer Theorie, die die Querschubverformung mitberücksichtigt, untersucht. Die erhaltenen Resultate werden mit den Vorhersagen der verbesserten Loveschen Theorie und der vereinfachten Donnellschen Theorie, welche keine Querschubverformung in Betrach ziehen, verglichen.Die durchgerechneten Beispiele zeigen, daß die Querschubverformung nicht nur für kurze, komposite Schalen sondern sogar für längere Schalen mit niederem Schubmodul Bedeutung gewinnen kann.

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Nomenclature a radius of reference surface of cylindrical shell - A _i amplitudes of displacements - A _ij elastic area - B _ij elastic statical moment - D _ij elastic moment of inertia - E _ij elastic stiffness modulus - h shell thickness - k ₄₄,k ₅₅ shear correction factors - l length of cylindrical shell - L _ij functional operator - m number of half-waves in axial direction - M ,M ,M axial, circumferential and twisting moments, respectively - n number of circumferential waves - N ⁰ axial compression force - N ,N ,N axial, circumferential and shear forces, respectively - Q ,Q transverse shear forces - R ₀,R ₁,R ₂ inertia terms - t time - T matrix defined in Eq. (25) - u, v, w displacements in axial, circumferential and radial directions, respectively - U _i time-independent displacement or rotation - nondimensional axial wave parameter - , , axial, circumferential and shear strains, respectively - radial coordinate, taken positive inward - polar angle - , , curvature changes - circular frequency - nondimensional axial coordinate - density - , , , , stress components - , rotations of normal to undeformed midsurface With 6 Figures     

Nonlinear vibration and instability analysis of functionally graded CNT-reinforced cylindrical shells conveying viscous fluid resting on orthotropic Pasternak medium

In this study, nonlinear vibration and instability of embedded temperature-dependent cylindrical shell conveying viscous fluid resting on temperature-dependent orthotropic Pasternak medium are investigated. The equivalent material properties of nanocomposites are estimated using rule of mixture. Both cases of uniform distribution and functionally graded distribution patterns of reinforcements are considered. Based on orthotropic Mindlin shell theory, the governing equations are derived. Generalized differential quadrature method is applied for obtaining the frequency and critical fluid velocity of a system. The effects of different parameters, such as distribution type of single-walled carbon nanotubes (SWCNTs), volume fractions of SWCNTs, and Pasternak medium are discussed.     

Vibration analysis of orthotropic cantilever cylindrical shells with axial thickness variation

A semi-analytical finite element method is used to determine the free vibration characteristics of thin orthotropic cantilever circular cylindrical shells. Love's first approximation shell theory is used for the formulation. The effect of the variation of thickness along the axial direction on natural frequencies, especially on the lowest frequency, has been studied with a constraint on the total mass of the shell for a particular length to middle surface radius ratio. Studies have been conducted for different degrees of orthotropy and various values of length to radius ratio.     

Free vibration of specially orthotropic,multilayered, thin cylindrical shells with various end conditions

The dynamic response of multilayered, specially orthotropic, thin, circular cylindrical shells is analysed using Love's first-approximation shell theory by the use of Rayleigh-Ritz variational procedure. A powerful tool is developed to calculate the natural frequencies for a variety of shell end conditions—clamped-free, clamped-supported, clamped-clamped, etc.—unlike most existing works, which are confined mainly to simply supported end conditions. The influence of various boundary conditions on the modal behaviour of a cylindrical shell is studied, and natural frequency calculations are established for a wide domain of shell geometries, using an appropriate dimensionless shell-geometrical parameter. The results of the present analysis are shown to compare very well with some available experimental results, in the case of an isotropic cylinder. Similar comparisons are not possible for the orthotropic case because of the nonavailability of previous results. Graphical illustrations of the results corresponding to the cases of various shell-geometrical and material parameters are provided for various end conditions.     

Buckling of moderately thick laminated cylindrical shells: a review

The present paper is a review article on the problem of buckling of moderately thick, laminated, composite shells subjected to destabilizing loads. The loads consist of uniform axial compression, uniform lateral pressure and torsion applied individually or in combination. In all the works reported in the literature, the analysis is based on higher-order shear deformation (HOSD) shell theory and/or first-order shear deformation (FOSD) shell theory with or without a shear correction factor. Results obtained by these two shell theories and by employing classical thin shell theory are compared to determine the range of applicability of each in predicting critical conditions. The effect of stacking sequence, radius-to-thickness ratio and length-to-radius ratio is assessed. Typical numerical results are presented in tabular form. Moreover, some limited results, which are based on limit point analysis are also presented (imperfection sensitivity studies).     

Free vibration of symmetric angle-ply thick laminated composite cylindrical shells

Analytical solutions for the vibrations of thick symmetric angle-ply laminated composite cylindrical shells are established using the first-order shear deformation theory. The complex method is developed to deal with the partial differential governing equations of thick symmetric angle-ply laminated composite cylindrical shells. Numerical results are provided for comparison, and coincide with existing results in the literature. The frequency characteristics for thick symmetric angle-ply laminated composite cylindrical shells with different H/R and L/R ratios are studied in comparison with those of symmetric cross-ply laminates. Also, the influence of lamination angle and number of lamination layers on frequency is investigated in details.     

Three-dimensional elasticity solution for static response of simply supported orthotropic cylindrical shells

This paper deals with the analysis of homogeneous and laminated cylindrical panels made of an orthotropic material, such as the fiber reinforced composites, using the three-dimensional elasticity equations. Solution is obtained by utilizing the assumption that the ratio of the panel thickness to its middle surface radius is negligible as compared to unity. However, it is shown that by sub-dividing the panel thickness into sub-layers of smaller thickness and matching the interface displacement and stress continuity conditions, very accurate results can be obtained. The two-dimensional shell theories have been compared for their accuracy in the light of the present three-dimensional elasticity analysis. Numerical results for some orthotropic panels show that the two-dimensional shell theories are very inaccurate when the thickness to length ratio of the panel is more than 1/20. Also, it is observed that the predictions of the two-dimensional shell theories are relatively poor in the case of two-layered panels as compared to three-layered and homogeneous panels.     

Vibrations and buckling of composite orthotropic cylindrical shells with nonuniform axial loads

The vibrational response of orthotropic composite cylindrical shells, subjected to circumferentially nonuniform axial loads, is investigated based on Flügge-type field equations. The use of a complex finite Fourier transform provides a simple method for handling any arbitrary nonuniform load but introduces modal coupling between the transformed equations. For simply supported boundaries (conditions SS3) the determination of the critical buckling load reduces to finding the eigenvalues of a finite matrix. Two different nonuniform loads are considered, having forms proportional to (1+2cos θ) and (θ*−θ), where is the Heaviside function, θ is the circumferential coordinate and aθ* is the width of an axial strip of the shell of radius a. Computed results indicate the sensitivity of the critical buckling loads and free vibrational frequency to the type of nonuniform load and the material lay-ups of the cylinders.     

Nonlinear dynamic buckling of orthotropic cylindrical shells subjected to rapidly applied loads

Dynamic buckling of an orthotropic cylindrical shell which is subjected to rapidly applied compression is considered. A nonlinear differential equation of Donnell–Karman type is derived with the initial imperfection taken into account. An energy method is used to obtain the equation of motion which is then solved numerically by means of a Runge-Kutta method. These numerical results show that the critical load is increased over the corresponding static case. An analytical solution is also obtained for the problem of hydrostatic pressure.