Effects of shear stresses and rotary inertia on the stability and vibration of sandwich cylindrical shells with FGM core surrounded by elastic medium

The vibration and stability of axially loaded sandwich cylindrical shells with the functionally graded (FG) core with and without shear stresses and rotary inertia resting Pasternak foundation are investigated. The dynamic stability is derived based on the first order shear deformation theory (FSDT) including shear stresses. The axial load and dimensionless fundamental frequency for FG sandwich shell with shear stresses and rotary inertia and resting on the Pasternak foundation. Finally, the influences of variations of FG core, elastic foundations, shear stresses and rotary inertia on the fundamental frequencies and critical axial loads are investigated.     

The nonlinear vibrations of functionally graded cylindrical shells surrounded by an elastic foundation

This paper reports the result of an investigation on the nonlinear vibrations of functionally graded cylindrical shell surrounded by an elastic foundation, based on Hamilton’s principle, von Kármán nonlinear theory, and the first-order shear deformation theory. Material properties are assumed to be temperature dependent. The surrounding elastic medium is modeled as Winkler foundation model, Pasternak foundation model, and nonlinear foundation model. Galerkin’s method is utilized to convert the governing partial differential equations to nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Considering the primary resonance case, the method of multiple scales is used to study the frequency response of nonlinear vibrations and the softening/hardening behavior. Parametric effects on the nonlinear vibrations are investigated.     

Postbuckling of FGM cylindrical shells under combined axial and radial mechanical loads in thermal environments

A postbuckling analysis is presented for a shear deformable functionally graded cylindrical shell of finite length subjected to combined axial and radial loads in thermal environments. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the shell surface and varied in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The formulations are based on a higher order shear deformation shell theory with von Kármán–Donnell-type of kinematic nonlinearity. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of functionally graded cylindrical shells. A singular perturbation technique is employed to determine the interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect cylindrical shells with two constituent materials subjected to combined axial and radial mechanical loads and under different sets of thermal environments. The results reveal that the temperature field and volume fraction distribution have a significant effect on the postbuckling behavior, but they have a small effect on the imperfection sensitivity of the functionally graded shell.     

On the theory of loaded general cylindrical Cosserat elastic shells

We study the equilibrium of cylindrical Cosserat elastic shells under the action of body loads and tractions and couples distributed along its edges. The shells have arbitrary open or closed cross-sections and are made from an isotropic and homogeneous material. On the end edges, the appropriate resultant forces and resultant moments are prescribed. We consider the problem of Almansi for cylindrical Cosserat shells and obtain a solution expressed in the form of the displacement field.     

Nonlinear deformation and stability of reinforced elliptical cylindrical shells loaded by internal pressure under twisting and bending

The problem of stability of cylindrical shells with an elliptical cross-sectional contour reinforced by a set of stringers under combined loading by bending and twisting moments, transverse force, and internal pressure is studied with the use of the variational method of finite elements in displacements. The subcritical stress-strain state of the shells is assumed to be moment and nonlinear. The effect of nonlinearity of deformation of the shells and their ellipticity on the critical loads and buckling type is determined.     

Postbuckling of imperfect stiffened cylindrical shells under combined external pressure and heating

1.IntroductionStiffenedcylindricalshellsarewidelyusedinmanytypesofstructures.Inpracticetheyoftensubjecttovarioustypesofcombinedthermalandmechanicalloadingandmayhavesignificantandunavoidableinitialgeometricalimperfections.Therefore,thepostbucklingbehaviorofimperfectstiITenedcylindricalshellsundercombinedexternalpress.ureandthermalloadingmustbewellunderstood.Manypostbucklingstudieshavebeenmadetbrstiffenedcylindricalshellsunderpureaxialcompression,uniformexternalpressureortheircombinations,where…     

Vibration analysis of bi-layered FGM cylindrical shells

In the present study, a vibration frequency analysis of a bi-layered cylindrical shell composed of two independent functionally graded layers is presented. The thickness of the shell layers is assumed to be equal and constant. Material properties of the constituents of bi-layered functionally graded cylindrical shell are assumed to vary smoothly and continuously through the thickness of the layers of the shell and are controlled by volume fraction power law distribution. The expressions for strain–displacement and curvature–displacement relationships are utilized from Love’s first approximation linear thin shell theory. The versatile Rayleigh–Ritz approach is employed to formulate the frequency equations in the form of eigenvalue problem. Influence of material distribution in the two functionally graded layers of the cylindrical shells is investigated on shell natural frequencies for various shell parameters with simply supported end conditions. To check the validity, accuracy and efficiency of the present methodology, results obtained are compared with those available in the literature.     

Postbuckling behavior of circular cylindrical shells under torsion

Summary Applying the Galerkin procedure to the Donnell basic equations, fairly accurate solutions are obtained for the postbuckling behavior of clamped circular cylindrical shells subjected to torsion. Through detailed calculations, main postbuckling characteristics, i.e., connections of the angle of twist, axial shortening and deflection with applied torque, are clarified for a wide range of shell geometrics. The results here obtained are ascertained to be in good agreement with the corresponding experimental results.

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Übersicht Durch Anwendung des Verfahrens von Galerkin auf die Donnellschen Grundgleichungen werden praktisch genaue Lösungen für das Nachbeulverhalten von eingespannten Kreiszylinderschalen gewonnen, die einer Torsion unterworfen sind. Über detaillierte Rechnungen für einen weiten Bereich von Schalengeometrien wird geklärt, wie Verdrehung, Stauchung und Nachbeulverformung mit dem gegebenen Torsionsmoment zusammenhängen. Die Ergebnisse stimmen mit experimentell ermittelten Werten gut überein.

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The authors wish to express their thanks to Mr. T. Watanabe at the Sumitomo Heavy Machinery, Ltd., and Mr. T. Sato at the Institute, for their help in computations and drawings.     

Postbuckling behavior of circular cylindrical shells under compression

Applying the Galerkin procedure to the Donnell basic equations, reasonably accurate solutions are obtained for the postbuckling behavior of clamped circular cylindrical shells under axial compression. To make a distinct comparison with the previous experimental results, calculations are carried out for shells with the same elastic and geometric parameters and the relations between the waveform, axial shortening and maximum deflections with applied loads are clarified. The results here obtained are found to be in reasonable agreement with experimental ones throughout the regions with fairly large deformations.     

Stability analysis of loaded coaxial cylindrical shells with internal fluid flow

We present numerical results for dynamical stability of loaded coaxial shells of revolution interacting with the internal fluid flow. The motion of the incompressible fluid is described in the framework of the theory of frictionless potential flow, whereas the static load acting on the shells is caused by the steady forces of viscous drag arising in the viscous turbulent flow in a closed channel. For shells with different boundary conditions, we study how the stability boundary is affected by the value of the gap between the shells for different versions of the outer shell rigidity and fluid flow. We show that, as in the case of unloaded coaxial shells, there is a significant deviation from the previous numerical and analytical results.     

Vibrations and buckling of axially loaded stiffened cylindrical shells with elastic restraints

A theory is derived for calculation of the influence of elastic edge restraints on the vibrations and buckling of stiffened cylindrical shells. The stiffeners are considered “smeared” and the edge restraints can be axial, radial, circumferential or rotational. Extensive computations are performed for special kinds of stringer-stiffened shells, and the theoretical predictions are compared with experimental results. A method of definition of equivalent elastically restrained boundary conditions by use of vibration tests is discussed. Application of this technique to tests on 10 shells significantly reduces the scatter in the ratio of experimental to predicted buckling loads.     

Compliance of cylindrical elastic shells reinforced by fibers

Kazan Institute of Chemical Engineering. Translated from Prikladnaya Mekhanika, Vol. 26, No. 2, pp. 71–75, February, 1990.     

Characteristics of elastic waves in FGM spherical shells,an analytical solution

In this paper, we investigate the propagation characteristics of elastic guided waves in FGM spherical shells with exponentially graded material in the radial direction. A new separation of variables technique to displacements is proposed to convert the governing equations of the wave motion to the second-order ordinary differential equations with variable coefficients. By further a variables transform technique, these equations are transformed to the Whittaker equations so that analytic solutions can be obtained. For the spherical shell case, by satisfying the traction-free boundary conditions on both the inner and outer surfaces of the shell, we obtain the dispersion equations, which show that both the SH and Lamb-type wave modes are generated in the structure. The calculated dispersion curves in the functionally graded shell demonstrate a clear influence of the gradient coefficient as compared to those of the homogeneous shell, with the Lamb-type waves more sensitive to the gradient coefficient. The mode shapes and the distributions of stresses in the shells for various gradient coefficients are also presented to illustrate their dependence on the gradient coefficient.     

Laterally loaded elastic shells of revolution

Summary An alternate form of the Chernin type equations for laterally loaded shells of revolution is obtained. Except for terms of order of the inherent error in shell theory, the final two coupled second order ordinary differential equations are remarkably similar to the Reissner-Meissner type equations for problems involving axi-symmetric stress distributions. Unlike all previous versions, our two equations can be further reduced (just as in the case of axi-symmetric stress distributions) to a single second order equation for a complex stress function without any additional approximation for uniform cylindrical, spherical, conical and toroidal shells. The side force and tilting moment problem for a shell frustum is shown to be the static geometric analogue of the problem of asymmetric bending and twisting of a ring shell sector. An efficient method for the evaluation of the overall influence coefficients is discussed. The stress state of a complete uniform spherical shell subject to concentrated side force and tilting moment at the two poles is analyzed.

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Übersicht Es wird eine neue Form der Cherninschen Gleichung für rotationssymmetrische, seitlich belastete Schalen abgeleitet. Abgesehen von Gliedern, deren Größenordnung dem allgemeinen Fehler der Schalentheorie entspricht, haben die zwei gekoppelten gewöhnlichen Differentialgleichungen zweiter Ordnung eine merkwürdige Ähnlichkeit zu den von Reissner und Meissner abgeleiteten Gleichungen für Probleme mit axialsymmetrischem Spannungszustand. Zum Unterschied zu früheren Fassungen der Cherninschen Gleichungen können die jetzigen zu einer einzigen Gleichung zweiter Ordnung auf eine komplexe Spannungsfunktion zusammengefaßt werden, wie dies im Falle rotationssymmetrischer Spannung auch möglich ist. Dabei sind keine zusätzlichen Näherungen für gleichförmige Zylinder-, Kugel-, Kegel- und Ringflächenschalen notwendig. Es zeigt sich, daß das Problem der Beanspruchung einer Stumpfschale durch Seitenkraft und Momente statisch und geometrisch analog ist zum Problem der Beanspruchung eines Ringschalensektors durch unsymmetrische Biegung und Drillung. Es wird ein Verfahren zur Bestimmung der Einflußkoeffizienten angegeben. Außerdem wird der Spannungszustand in einer gleichförmigen vollständigen Kugelschale analysiert, die durch eine einzelne Seitenkraft und Momente an den zwei Polen beansprucht wird.

     

Postbuckling analysis of axially-loaded laminated cylindrical shells with piezoelectric actuators

A compressive postbuckling analysis is presented for a laminated cylindrical shell with piezoelectric actuators subjected to the combined action of mechanical, electric and thermal loads. The temperature field considered is assumed to be a uniform distribution over the shell surface and through the shell thickness, and the electric field is assumed to be the transverse component E_Z only. The material properties are assumed to be independent of the temperature and the electric field. The governing equations are based on the classical shell theory with von Kármán–Donnell-type kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of hybrid laminated cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the compressive postbuckling behavior of perfect and imperfect, cross-ply laminated cylindrical thin shells with fully covered or embedded piezoelectric actuators under different sets of thermal and electric loading conditions. The effects played by temperature rise, applied voltage, shell geometric parameter, stacking sequence, as well as initial geometric imperfections are studied.     

Deformation of orthotropic noncircular cylindrical shells in an elastic bed

Institute of Mechanics, Academy of Sciences of the Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 29, No. 3, pp. 43–47, March, 1993.