Natural frequency analysis of functionally graded material truncated conical shell with lengthwise material variation based on first-order shear deformation theory

Based on the first-order shear deformation theory, the free vibration of the functionally graded (FG) truncated conical shells is analyzed. The truncated conical shell materials are assumed to be isotropic and inhomogeneous in the longitudinal direction. The two-constituent FG shell consists of ceramic and metal. These constituents are graded through the length, from one end of the shell to the other end. Using Hamilton's principle the derived governing equations are solved using differential quadrature method. Fast rate of convergence of this method is tested and its advantages over other existing solver methods are observed. The primary results of this study were obtained for four different end boundary conditions, and for some special cases, acquired results were compared with those available in the literature. Furthermore, effects of geometrical parameters, material graded power index, and boundary conditions on the natural frequencies of the FG truncated conical shell are carried out.     

Mechanical buckling of functionally graded material cylindrical shells surrounded by Pasternak elastic foundation

In this study, the mechanical buckling of functionally graded material cylindrical shell that is embedded in an outer elastic medium and subjected to combined axial and radial compressive loads is investigated. The material properties are assumed to vary smoothly through the shell thickness according to a power law distribution of the volume fraction of constituent materials. Theoretical formulations are presented based on a higher-order shear deformation shell theory (HSDT) considering the transverse shear strains. Using the nonlinear strain–displacement relations of FGMs cylindrical shells, the governing equations are derived. The elastic foundation is modelled by two parameters Pasternak model, which is obtained by adding a shear layer to the Winkler model. The boundary condition is considered to be simply-supported. The novelty of the present work is to achieve the closed-form solutions for the critical mechanical buckling loads of the FGM cylindrical shells surrounded by elastic medium. The effects of shell geometry, the volume fraction exponent, and the foundation parameters on the critical buckling load are investigated. The numerical results reveal that the elastic foundation has significant effect on the critical buckling load.     

Mechanical buckling of two-dimensional functionally graded cylindrical shells surrounded by Winkler–Pasternak elastic foundation

In this research, buckling analysis of a two-dimensional, functionally graded, cylindrical shell that has been embedded in an outer elastic medium in the presence of combined axial and transverse loading based on third-order shear deformation shell theory is numerically investigated. Variations of the shell properties are considered to be continuous through length and thickness. Winkler–Pasternak foundation and simply supported boundary conditions have been applied. The problem has been solved using the generalized differential quadrature method. Geometrical, load, and foundation parameters beside functionally graded power indexes effects on the critical buckling load have been studied.     

The buckling of FGM truncated conical shells subjected to combined axial tension and hydrostatic pressure

The purpose of this paper is to investigate the elastic buckling of FGM truncated thin conical shells under combined axial tension and hydrostatic pressure. Here axial tensions are separately applied to small and large bases of the truncated conical shell, respectively. It is assumed that the cone is a mixture of metal and ceramic, and that its properties changes as the power and exponential functions of the shell thickness. After giving the fundamental relations, the stability and compatibility equations of an FGM truncated conical shell, subject to combined axial tension and hydrostatic pressure, have been derived. Applying Galerkin’s method general formulas have been obtained for the critical combined and separate loads of FGM conical shells. The appropriate formulas for homogenous and FGM cylindrical shells are found as a special case. Effects of changing shell characteristics, material composition and volume fraction of constituent materials on the critical combined and separate loads of FGM shells with simply supported edges are also investigated. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

The nonlinear stability of axisymmetric functionally graded material annular spherical shells under thermo-mechanical load

The authors of this article investigate the nonlinear stability of axisymmetric functionally graded annular spherical shells with temperature-dependent material properties subjected to thermo-mechanical loads and resting on elastic foundations. Equilibrium and compatibility equations are derived by using the classical thin shell theory in terms of the shell deflection and the stress function. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to obtain the closed-form of load–deflection paths. An analysis is carried out to show the effects of material and geometrical properties and combination of loads on the stability of the shells.     

Free vibration and flexural response of functionally graded plates resting on Winkler–Pasternak elastic foundations using nonpolynomial higher-order shear and normal deformation theory

In the present work, the flexural and vibration response of a functionally graded plate resting on Pasternak elastic foundation is analyzed using a recently developed nonpolynomial higher-order shear and normal deformation theory by the authors. The novelty of this theory is that it contains only four unknowns and also accommodates the thickness stretching effect. Two kinds of micromechanics models, namely, the Voigt and Mori–Tanaka models, are considered. Material properties of the functionally graded plates are assumed to vary continuously in the thickness direction according to either a simple power law or an exponential law. Finite element formulation is done using C° continuous Lagrangian quadrilateral nine-noded elements with eight degrees of freedom per node. The equations of motion are derived using a variational approach. Convergence and comparison studies are carried out to establish the authenticity and reliability of the solutions. The effect of various boundary conditions, geometric conditions, micromechanics models, and foundation parameters on the flexural and vibration response of the functionally graded plate are investigated.     

Buckling analysis of moderately thick FG carbon nanotube reinforced composite conical shells under axial compression by DQM

A semi-analytical method is presented to study the buckling of moderately thick carbon nanotube reinforced composite conical shells under axial compression. Material properties are assumed to be graded across the shell thickness. In order to derive the equilibrium equations of the shell, first-order shear deformation theory is used. Then, the partial differential equations are transformed to algebraic type by using the differential quadrature method. To validate the results obtained in this study, comparisons are made with the outcomes of previous studies. Finally, the effects of the geometry of the shell, circumferential mode number, volume fraction of the carbon nanotube, and boundary conditions are studied.     

Nonlinear dynamic response and vibration of imperfect shear deformable functionally graded plates subjected to blast and thermal loads

Based on Reddy's higher-order shear deformation plate theory, this article presents an analysis of the nonlinear dynamic response and vibration of imperfect functionally graded material (FGM) thick plates subjected to blast and thermal loads resting on elastic foundations. The 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. Numerical results for the dynamic response and vibration of the FGM plates with two cases of boundary conditions are obtained by the Galerkin method and fourth-order Runge–Kutta method. The results show the effects of geometrical parameters, material properties, imperfections, temperature increment, elastic foundations, and boundary conditions on the nonlinear dynamic response and vibration of FGM plates.     

Experimental investigations on buckling of cylindrical shells under axial compression and transverse shear

This paper presents experimental studies on buckling of cylindrical shell models under axial and transverse shear loads. Tests are carried out using an experimental facility specially designed, fabricated and installed, with provision forin-situ measurement of the initial geometric imperfections. The shell models are made by rolling and seam welding process and hence are expected to have imperfections more or less of a kind similar to that of real shell structures. The present work thus differs from most of the earlier investigations. The measured maximum imperfections δ_max are of the order of ±3t (t = thickness). The buckling loads obtained experimentally are compared with the numerical buckling values obtained through finite element method (FEM). In the case of axial buckling, the imperfect geometry is obtained in four ways and in the case of transverse shear buckling, the FE modelling of imperfect geometry is done in two ways. The initial geometric imperfections affect the load carrying capacity. The load reduction is considerable in the case of axial compression and is marginal in the case of transverse shear buckling. Comparisons between experimental buckling loads under axial compression, reveal that the extent of imperfection, rather than its maximum value, in a specimen influences the failure load. Buckling tests under transverse shear are conducted with and without axial constraints. While differences in experimental loads are seen to exist between the two conditions, the numerical values are almost equal. The buckling modes are different, and the experimentally observed and numerically predicted values are in complete disagreement.     

Thermal post-buckling analysis of functionally graded material elliptical plates based on high-order shear deformation theory

Thermal post-buckling analysis is first presented for functionally graded elliptical plates based on high-order shear deformation theory in different thermal environments. Material properties are assumed to be temperature-dependent and graded in the thickness direction. Ritz method is employed to determine the central deflection-temperature curves, the validity of which can be confirmed by comparison with related researchers' results; it is worth noting that the forms of approximate solutions are well chosen in consideration of both simplicity and accuracy. Influences played by different supported boundaries, thermal environmental conditions, ratio of major to minor axis, and volume fraction index are discussed in detail.     

Analytical investigation on the free vibration behavior of rotating FGM truncated conical shells reinforced by orthogonal eccentric stiffeners

This article presents an analytical investigation on the free vibration behavior of rotating functionally graded truncated conical shells reinforced by stringers and rings with the change of spacing between stringers. Using the Donnell shell theory, smeared stiffeners technique, and taking into account the influences of centrifugal force and Coriolis acceleration, the governing equations are derived. These variable coefficient partial differential equations are studied by the Galerkin method. The sixth-order polynomial equation of natural frequency is obtained. Numerical results show effects of stiffener and input parameters on the frequency of shell.     

Thermal effects on nonlinear vibrations of functionally graded doubly curved shells using higher order shear deformation theory

Geometrically nonlinear vibrations of functionally graded (FG) doubly curved shells subjected to thermal variations and harmonic excitation are investigated via multi-modal energy approach. Two different nonlinear higher-order shear deformation theories are considered and it is assumed that the shell is simply supported with movable edges. Using Lagrange equations of motion, the energy functional is reduced to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities which is truncated based on solution convergence. A pseudo-arclength continuation and collocation scheme is employed to obtain numerical solutions for shells subjected to static and harmonic loads. The effects of FGM power law index, thickness ratio and temperature variations on the frequency–amplitude nonlinear response are fully discussed and it is revealed that, for relatively thick and deep shells, the Amabili–Reddy theory which retains all the nonlinear terms in the in-plane displacements gives different and more accurate results.     

On the linear stability of eccentrically stiffened functionally graded annular spherical shell on elastic foundations

The study deals with the formulation of governing equations of eccentrically stiffened functionally graded materials annular spherical shells resting on elastic foundations and based upon the classical shell theory and the smeared stiffeners technique taking into account geometrical nonlinearity in Von Karman-Donnell sense. The annular spherical shells are reinforced by eccentrically longitudinal and transversal stiffeners made of full metal or full ceramic depending on situation of stiffeners at metal-rich side or ceramic-rich side of the shell respectively. Approximate solutions are assumed to satisfy the simply supported boundary condition and Galerkin method is applied to obtain closed-form relations of bifurcation type of buckling loads. Numerical results are given to evaluate effects of inhomogeneous, dimensional parameters, outside stiffeners and elastic foundations to the buckling of structures.     

Influences of elastic foundations and boundary conditions on the buckling of laminated shell structures subjected to combined loads

This article presents to study the stability of laminated orthotropic cylindrical and truncated conical shells resting on elastic foundations and subjected to combined loads with the clamped and simply supported boundary conditions. Here, axial tensile loads separately applied to the small and large bases of a laminated truncated conical shell, respectively. The basic relations, the modified Donnell type stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells on the Pasternak type elastic foundation. Applying Galerkin method, the critical combined loads of laminated orthotropic conical shells on the Pasternak type elastic foundation with different boundary conditions are obtained. The appropriate formulas for single-layer and laminated cylindrical shells on the Pasternak type elastic foundation made of orthotropic and isotropic materials are found as special cases. Finally, influences of the boundary conditions, the elastic foundation, the number and ordering of the layers and variations of the shell characteristics on the critical combined loads are investigated. The results are compared with their counterparts in the literature.     

Nonlinear bending of FGM plates subjected to combined loading and resting on elastic foundations

A nonlinear bending analysis is presented for a simply supported, functionally graded plate resting on an elastic foundation of Pasternak-type. The plate is exposed to elevated temperature and is subjected to a transverse uniform or sinusoidal load combined with initial compressive edge loads. 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 plate theory and general von Kármán-type equation that includes the plate-foundation interaction and thermal effects. A two step perturbation technique is employed to determine the load–deflection and load–bending moment curves. The numerical illustrations concern nonlinear bending response of functional graded plates with two constituent materials resting on Pasternak elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The results reveal that the characteristics of nonlinear bending are significantly influenced by foundation stiffness, temperature rise, transverse shear deformation, the character of in-plane boundary conditions and the amount of initial compressive load. In contrast, the effect of volume fraction index N becomes weaker when the plate is supported by an elastic foundation.     

Vibration and buckling analysis of two-layered functionally graded cylindrical shell,considering the effects of transverse shear and rotary inertia

This research investigates the free vibration and buckling of a two-layered cylindrical shell made of inner functionally graded (FG) and outer isotropic elastic layer, subjected to combined static and periodic axial forces. Material properties of functionally graded cylindrical shell are considered as temperature dependent and graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. Theoretical formulations are presented based on two different methods of first-order shear deformation theory (FSDT) considering the transverse shear strains and the rotary inertias and the classical shell theory (CST). The results obtained show that the transverse shear and rotary inertias have considerable effect on the fundamental frequency of the FG cylindrical shell. The results for nondimensional natural frequency are in a close agreement with those in literature. It is inferred from the results that the geometry parameters and material composition of the shell have significant effect on the critical axial force, so that the minimum critical load is obtained for fully metal shell. Good agreement between theoretical and finite element results validates the approach. It is concluded that the presence of an additional elastic layer significantly increases the nondimensional natural frequency, the buckling resistance and hence the elastic stability in axial compression with respect to a FG hollow cylinder.     

Using the third-order shear deformation theory to examine magnetic-mechanical buckling of a two-dimensional functionally graded cylindrical panel with longitudinal and circumferential stiffeners

In this study, the magnetic-mechanical buckling of a cylindrical panel made of two-dimensional functionally graded materials (2D-FGMs) has been investigated. The panel contains longitudinal and circumferential stiffeners and has been subjected to a uniform magnetic field as well as axial load. Material properties of the cylindrical panel are assumed to vary continuously in radial and thickness directions as a function of the volume fraction of the components. The magnetic field has been exerted radially. Equilibrium and stability equations have been derived using both Hamilton's principle and principle of minimum potential energy based on the third-order shear deformation theory (TSDT). The generalized differential quadrature method (GDQ) has been employed to solve the coupled differential equations. Moreover, the effect of geometry, load, magnitude of the magnetic field, number of stiffeners, and volume fraction coefficient on the critical buckling load has been determined. The results are in good agreement with the previous related works.     

Mechanical and thermal postbuckling of higher order shear deformable functionally graded plates on elastic foundations

This paper presents an analytical investigation on the buckling and postbuckling behaviors of thick functionally graded plates resting on elastic foundations and subjected to in-plane compressive, thermal and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation plate theory taking into account Von Karman nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, closed-form relations of buckling loads and postbuckling equilibrium paths for simply supported plates are determined. Analysis is carried out to show the effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the plates.     

Evaluation of nonlocal higher order shear deformation models for the vibrational analysis of functionally graded nanostructures

Small scale effects in the functionally graded beam are investigated by using various nonlocal higher-order shear deformation beam theories. The material properties of a beam are supposed to vary according to power law distribution of the volume fraction of the constituents. The nonlocal equilibrium equations are obtained and an exact solution is presented for vibration analysis of functionally graded (FG) nanobeams. The accuracy of the present model is discussed by comparing the results with previous studies and a parametric investigation is presented to study the effects of power law index, small-scale parameter, and aspect ratio on the vibrational behavior of FG nanostructures.     

On the dynamic buckling of truncated conical shells with functionally graded coatings subject to a time dependent axial load in the large deformation

A study has been made to determine the critical time parameters of truncated conical shells with functionally graded coatings (FGCs) and subjected to a time dependent axial load in the large deformation. The method of solution utilizes Superposition principle and Galerkin procedure. Donnell–Karman type non-linear differential equations for the truncated conical shell with FGCs are derived and reduced to ordinary differential equation with the time dependent coefficient. The Runge–Kutta method and modified Budiansky–Roth criterion are then used to solve this non-linear differential equation with the time dependent coefficient. Finally, effects of compositional profiles of coatings, variation of truncated conical shell parameters and loading speed on the dimensionless linear and non-linear critical time parameters have been studied. Comparing the results of this study with those in the literature validates the present analysis.