A.H. Sofiyev

The buckling of an orthotropic composite truncated conical shell with continuously varying thickness subject to a time dependent external pressure

In this study, the buckling of an orthotropic composite truncated conical shell with continuously varying thickness, subject to a uniform external pressure which is a power function of time, has been considered. At first, the fundamental relations and the Donnell type stability equations of an orthotropic composite truncated conical shell, subject to an external pressure, have been obtained. Then, employing Galerkin method, those equations have been reduced of time dependent differential equation with variable coefficients. Finally, applying the variational method of Ritz method type, the critical static and dynamic loads, the corresponding wave numbers and the dynamic factor have been found analytically. Using those results, the effects of the variations of the power in the thickness expression, the semi-vertex angle, the power of time in the external pressure expression and the ratio of the Youngs moduli on the critical parameters are studied numerically, for the case when the thickness of the conical shell varies as a power and exponential function. It is observed, from the computations carried out, that these factors have appreciable effects on the critical parameters of the problem in the heading.

Stability and vibration of sandwich cylindrical shells containing a functionally graded material core with transverse shear stresses and rotary inertia effects

The dimensionless fundamental frequencies and critical axial loads of sandwich cylindrical thin shell with a functionally graded (FG) core are studied by taking into account the combined and separately influences of the shear stresses and rotary inertia. The governing equations of sandwich cylindrical shell with an FG core are derived based on Donnell’s shell theory using the shear deformation theory. The governing equations are reduced the sixth-order algebraic equation using the Galerkin’s method. Numerically solving this algebraic equation gives the magnitudes of the dimensionless fundamental frequency. In addition, the expressions for the dimensionless fundamental frequencies and critical axial loads of the sandwich cylindrical shell containing an FG core with and without the shear stresses are obtained in a special case. To validate the present method, the numerical example is presented and compared with the available existing results. Finally, the influences of variations of the FG core, shear stresses, rotary inertia and sandwich shell geometry parameters on the dimensionless fundamental frequencies and critical axial loads are analyzed numerically.

The Vibration Analysis of FGM Truncated Conical Shells Resting on Two-Parameter Elastic Foundations

An analytical formulation is presented for the vibration analysis of truncated conical shells made of functionally graded material and resting on the Winkler-Pasternak foundations. It is assumed that the truncated conical shell is a mixture of metal and ceramic that its properties changes as a function of the shell thickness. The governing equations according to the Donnells theory are solved by Galerkins method and the fundamental frequencies with or without two-parameter elastic foundation have been found. The effects of the elastic foundations, changing large radius-to-small radius ratio, lengths-to-radius ratio, material composition and volume fraction of constituent materials on the fundamental frequencies of the truncated conical shell are investigated.

Effects of elastic foundation on the vibration of laminated non-homogeneous orthotropic circular cylindrical shells

In this paper an analytical procedure is given to study the free vibration characteristics of laminated non-homogeneous orthotropic thin circular cylindrical shells resting on elastic foundation, accounting for Karman type geometric non-linearity. At first, the basic relations and modified Donnell type stability equations, considering finite deformations, have been obtained for laminated thin orthotropic circular cylindrical shells, the Youngs moduli of which varies piecewise continuously in the thickness direction. Applying Galerkin method to the latter equations, a non-linear time dependent differential equation is obtained for the displacement amplitude. The frequency is obtained from this equation as a function of the shell displacement amplitude. Finally, the effect of elastic foundation, non-linearity, non-homogeneity, the number and ordering of layers on the frequency is found for different mode numbers. These results are given in the form of tables and figures. The present analysis is validated by comparing results with those in the literature.

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

ABSTRACT 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.

Vibration and stability of axially compressed truncated conical shells with functionally graded middle layer surrounded by elastic medium

The vibration and stability analyses are presented for axially compressed three-layered truncated conical shells with a functionally graded (FG) middle layer surrounded by elastic media. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to simple power law and exponential distributions in terms of the volume fractions of the constituents. Five sets of the material mixture are considered. The Pasternak model is used to describe the reaction of the elastic medium on the truncated conical shell. The fundamental relations, the modified Donnell-type dynamic stability and compatibility equations for the three-layered truncated conical shell with an FGM middle layer are derived. The governing equations are solved by using the Galerkin method and obtained expressions for dimensionless frequency parameters and dimensionless critical axial loads for three-layered truncated conical shells with the FG middle layer with and without an elastic foundation. The numerical results reveal that variations of the shell thickness-to-FGM thickness ratio, lengths-to-radius ratio, Winkler foundation stiffness, shear subgrade modulus of the foundation, material mixture and compositional profiles of the FG middle layer have significant effects on the values of dimensionless critical axial load and dimensionless frequency parameter. The results are verified by comparing the obtained values with those in the existing literature.

EFFECT OF THE TWO-PARAMETER ELASTIC FOUNDATION ON THE CRITICAL PARAMETERS OF NONHOMOGENEOUS ORTHOTROPIC SHELLS

A theoretical analysis is presented for determining the free vibrational and buckling characteristics of the nonhomogeneous, orthotropic, thin-walled, circular cylindrical and conical shells under a hydrostatic pressure and resting on a two-parameter elastic foundation. The basic relations have been obtained for the orthotropic truncated conical shell, the Youngs moduli and density of which vary continuously in the thickness direction. By applying the Galerkin method, the buckling hydrostatic pressure and dimensionless frequency parameter of the homogeneous and nonhomogeneous orthotropic truncated conical shells with or without elastic foundations are obtained. Finally, the effects of the Winkler and Pasternak-type elastic foundations, the variations of shell characteristics, the effects of the nonhomogeneity and orthotropy on the critical parameters have been studied. The results are presented in tables, figures and compared with other works.

The Stability of a Three-Layered Composite Conical Shell Containing a FGM Layer Subjected to External Pressure

In this study, the stability of a generic three-layered truncated conical shell containing a functionally graded (FG) layer subjected to uniform external pressure is investigated. The material properties of the functionally graded layer are assumed to vary continuously through the thickness of the shell. The variation of the properties follows an arbitrary distribution in terms of the volume fractions of the constituents. The fundamental relations, the stability and compatibility equations of three-layered truncated conical shells containing a FG layer are obtained, first. Then, applying Galerkins method, the closed form solution for critical external pressure is obtained. The results show that the critical parameters are affected by the configurations of the constituent materials, the variations of the thickness of the FG layer and the variation of the shell geometry. Comparing the results with those in the literature validates the present analysis.

The Dynamic Stability of Orthotropic Cylindrical Shells with Non-homogenous Material Properties under Axial Compressive Load Varying as a Parabolic Function of Time

In this study, the dynamic stability problem of a cylindrical shell composed of non-homogeneous orthotropic materials with Young’s moduli and density varying continuously in the thickness direction under the effect of an axial compressive load varying with a parabolic function of time is considered. At first, the fundamental relations and the modified Donnell type dynamic stability equations of a non-homogeneous orthotropic cylindrical shell are set up. Applying the Galerkin method, first, and then the Ritz-type variational method, the closed-form solutions have been derived for the dynamic critical axial load and dynamic factor. Finally, carrying out some computations, the effects of the non-homogeneity of the orthotropy ratio and the axial loading parameter on the critical parameters have been studied. Comparing results with those in the literature validates the present analysis.

Large-amplitude vibration of the geometrically imperfect FGM truncated conical shell

In this study, the large-amplitude vibration of a functionally graded (FG) truncated conical shell with an initial geometric imperfection has been investigated using large deformation theory with a von Karman–Donnell type of kinematic nonlinearity. The material properties of an FG truncated conical shell are assumed to vary continuously through the thickness. The fundamental relations, the modified Donnell-type nonlinear motion, and compatibility equations of the FG truncated conical shell with an initial geometric imperfection are derived. The relation between nonlinear frequency parameters with the dimensionless amplitude of imperfect FG truncated conical shells is obtained. Finally, the influences of variations of the initial geometric imperfection, compositional profiles, and shell characteristics on the dimensionless nonlinear frequency parameter and frequency–amplitude relations are investigated. The present results are compared with the available data for a special case.