Nonlinear vibration and dynamic response of shear deformable imperfect functionally graded double-curved shallow shells resting on elastic foundations in thermal environments

In this article, nonlinear vibration and dynamic response of imperfect functionally graded materials (FGM) thick double-curved shallow shells resting on elastic foundations are investigated using Reddy's third-order shear deformation shell theory in thermal environments. 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 FGM shells are subjected to mechanical, damping, and thermal loads. The Galerkin method and fourth-order \hboxRunge–Kutta method are used to calculate natural frequencies, nonlinear frequency–amplitude relation, and dynamic response of the shells. In numerical results, the effects of geometrical parameters, material properties, imperfections, shear deformation, the elastic foundations, mechanical, thermal and damping loads on the nonlinear dynamic response, and nonlinear vibration of FGM double-curved shallow shells are investigated. Accuracy of the present formulation is shown by comparing the results of numerical examples with the ones available in literature.     

Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of harmonic differential quadrature-finite difference methods

The nonlinear dynamic response of doubly curved shallow shells resting on Winkler–Pasternak elastic foundation has been studied for step and sinusoidal loadings. Dynamic analogues of Von Karman–Donnel type shell equations are used. Clamped immovable and simply supported immovable boundary conditions are considered. The governing nonlinear partial differential equations of the shell are discretized in space and time domains using the harmonic differential quadrature (HDQ) and finite differences (FD) methods, respectively. The accuracy of the proposed HDQ-FD coupled methodology is demonstrated by numerical examples. The shear parameter G of the Pasternak foundation and the stiffness parameter K of the Winkler foundation have been found to have a significant influence on the dynamic response of the shell. It is concluded from the present study that the HDQ-FD methodolgy is a simple, efficient, and accurate method for the nonlinear analysis of doubly curved shallow shells resting on two-parameter elastic foundation.     

Thermal stability of eccentrically stiffened FGM plate on elastic foundation based on Reddy's third-order shear deformation plate theory

This article studies the nonlinear thermal buckling and postbuckling of eccentrically stiffened functionally graded plates on elastic foundation subjected to mechanical, thermal, and thermomechanical loads. The noticeable point of this study is using the Reddy's higher order shear deformation plate theory and a general formula for the forces and moments of eccentrically stiffened functionally graded material (FGM) plate, which takes into account the influence of temperature on both the FGM plate and stiffeners. The article used the Galerkin method, stress function, and iterative method to determine the thermal buckling loads and postbuckling response of the eccentrically stiffened FGM plates in three different cases of boundary conditions. The effects of material, temperature-dependent material properties, elastic foundations, boundary conditions, outside stiffeners, and temperature on the buckling and postbuckling loading capacity of the FGM plates in thermal environments are analyzed and discussed. A good agreement is obtained by comparing the present analysis with other available literature.     

Nonlinear dynamic response of eccentrically stiffened FGM plate using Reddy’s TSDT in thermal environment

The nonlinear dynamics of an eccentrically stiffened functionally graded material (ES-FGM) plates resting on the elastic Pasternak foundations subjected to mechanical and thermal loads is considered in this article. The plates are reinforced by outside stiffeners with temperature-dependent material properties in two cases: uniform temperature rise and through the thickness temperature gradient. Both stiffeners and plate are deformed under temperature. Using Reddy’s third-order shear deformation plate theory, stress function, Galerkin and fourth-order Runge–Kutta methods, the effects of material and geometrical properties, temperature-dependent material properties, elastic foundations, and stiffeners on the nonlinear dynamic response of the ES-FGM plate in thermal environments are studied and discussed. Some obtained results are validated by comparing with those in the literature.     

The buckling of FGM truncated conical shells subjected to axial compressive load and resting on Winkler–Pasternak foundations

In this study, the buckling analysis of the simply supported truncated conical shell made of functionally graded materials (FGMs) is presented. The FGM truncated conical shell subjected to an axial compressive load and resting on Winkler–Pasternak type elastic foundations. The material properties of functionally graded shells are assumed to vary continuously through the thickness. The modified Donnell type stability and compatibility equations are solved by Galerkin’s method and the critical axial load of FGM truncated conical shells with and without elastic foundations have been found analytically. The appropriate formulas for homogenous and FGM cylindrical shells with and without elastic foundations are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation. Finally, parametric studies on the buckling of FGM truncated conical and cylindrical shells on elastic foundations are being investigated. These parameters include; power-law and exponential distributions of FGM, Winkler foundation modulus, Pasternak foundation modulus and aspect ratios of shells.     

Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations

This paper investigates the nonlinear dynamic response of thick functionally graded materials (FGM) plates using the third-order shear deformation plate theory and stress function. The FGM plate is assumed to rest on elastic foundations and subjected to thermal and damping loads. Numerical results for dynamic response of the FGM plate are obtained by Runge–Kutta method. The results show the influences of geometrical parameters, the material properties, the elastic foundations, and thermal loads on the nonlinear dynamic response of FGM plates.     

Dynamic Thermal Postbuckling Analysis of Shear Deformable Piezoelectric-FGM Cylindrical Shells

Dynamic thermal postbuckling behavior of functionally graded cylindrical shells with surface-bonded piezoelectric actuators subjected to the combined action of thermal load and applied actuator voltage is studied. The shell material is graded across the thickness according to a power law. The material properties of the functionally graded cylindrical shells are considered to be temperature dependent. The theoretical formulations are based on the Sanders nonlinear kinematic relations, which account for the transverse shear strains, and the third-order shear deformation shell theory is employed. Hamilton's principle is used to derive the equations of motion governing piezoelectric FGM cylindrical shells. A finite difference approximation combined with the Runge-Kutta method is employed to predict the postbuckling equilibrium paths, and the dynamic buckling temperature difference is detected according to Budiansky's stability criterion. Numerical results are presented to demonstrate the effects of the applied actuator voltage, shell geometry, volume fraction exponent in the power-law variation of the FGM, and the temperature dependency of the material properties on the postbuckling behavior of the shell. The results for simpler states are validated with the known results in the literature.     

Dynamic Thermal Postbuckling Analysis of Piezoelectric Functionally Graded Cylindrical Shells

Dynamic thermal postbuckling behavior of functionally graded cylindrical shells with surface-bonded piezoelectric actuators subjected to the combined action of thermal load and applied actuator voltage is analyzed using an incremental numerical technique. The shell is graded across the thickness according to a power law form function. The material properties of the functionally graded cylindrical shells are considered to be temperature dependent. The theoretical formulations are based on the classical shell theory with Sanders' nonlinear kinematic relations. Then, using Hamilton's principle, equations of motion are derived for the piezoelectric FGM cylindrical shell. A finite difference based method combined with the Runge–Kutta method is employed to predict the postbuckling equilibrium paths, and the dynamic buckling temperature difference is detected according to Budiansky's stability criterion. Numerical results are presented to demonstrate the effects of the applied actuator voltage, shell geometry, volume fraction exponent of FGM, and the temperature dependency of the material properties on the postbuckling behavior of the shell. The results for simpler states are validated with the known data in the literature.     

Axisymmetric nonlinear rapid heating of FGM cylindrical shells

Large amplitude thermally induced vibrations of cylindrical shells made of a through-the-thickness functionally graded material (FGM) are investigated in the current research. All of the thermo-mechanical properties of the FGM shell are assumed to be functions of temperature and thickness coordinate. Shell is subjected to rapid surface heating on the ceramic-rich surface while the other surface of the shell is kept at reference temperature. One dimensional heat conduction equation is constructed and solved by means of a hybrid finite difference-Crank–Nicolson algorithm. The constructed heat conduction equation is nonlinear since the thermal conductivity is temperature dependent. With the aid of first-order shear deformation shell theory under the axisymmetric Donnell kinematic assumptions and von Kármán type of strain-displacement relations, the total energy of the shell is established. Implementing the conventional Ritz method, a set of nonlinear coupled algebraic equations are obtained which govern the dynamics of the shell under thermal shock. These equations are solved in time domain using the Newmark time marching scheme and the simple Picard successive method. Parametric studies are given to explore the dynamics of an FGM cylindrical shell under thermal shock.     

Thermal buckling analysis of FGM sandwich truncated conical shells reinforced by FGM stiffeners resting on elastic foundations using FSDT

This work presents an analytical approach to investigate the mechanical and thermal buckling of functionally graded materials sandwich truncated conical shells resting on Pasternak elastic foundations, subjected to thermal load and axial compressive load. Shells are reinforced by closely spaced stringers and rings, in which the material properties of shells and stiffeners are graded in the thickness direction following a general sigmoid law distribution and a general power law distribution. Four models of coated shell-stiffener arrangements are investigated. The change of spacing between stringers in the meridional direction also is taken into account. Two cases on uniform temperature rise and linear temperature distribution through the thickness of shell are considered. Using the first-order shear deformation theory, Lekhnitskii smeared stiffener technique and the adjacent equilibrium criterion, the linearization stability equations have been established. Approximate solution satisfies simply supported boundary conditions and Galerkin method is applied to obtain closed-form expression for determining the critical compression buckling load and thermal buckling load in cases uniform temperature rise and linear temperature distribution across the shell thickness. The effects of temperature, foundation, core layer, coating layer, stiffeners, material properties, dimensional parameters and semi-vertex angle on buckling behaviors of shell are shown.     

Unsymmetrical Buckling of Piezo-FGM Shallow Clamped Spherical Shells under Thermal Loading

The unsymmetrical buckling of clamped shallow spherical shells made of functionally graded material (FGM) and surface-bonded piezoelectric actuators under thermal load is studied in this paper. The governing equations are based on classical shell theory and the Sanders nonlinear kinematic equations. It is assumed that properties of the functionally graded material vary continuously through the thickness of the shell according to a power law distribution of the volume fractions of the constituent materials.     

Application of two-steps perturbation technique to geometrically nonlinear analysis of long FGM cylindrical panels on elastic foundation under thermal load

Present research deals with the geometrically nonlinear bending of a long cylindrical panel made of a through-the-thickness functionally graded material subjected to thermal load. A panel under the action of uniform temperature rise loading is considered. Formulation of the shell is based on the third-order shear deformation shell theory, where the first-order shear deformation and classical shell theory may be extracted as special cases. Thermomechanical properties of the shell are assumed to be temperature dependent and are estimated according to a power law function across the shell thickness. Also, it is assumed that shell is in contact with an elastic foundation which acts in tension as well as in compression. The nonlinear governing equations of the shell are obtained using the von Kármán type of geometrical nonlinearity. The obtained governing equations are solved for two cases, i.e., simply supported shells and clamped shells. The developed equations are solved using a two-step perturbation technique. Accurate closed-form expressions are provided to obtain the mid-span deflection of the shell as a function of temperature elevation. Numerical results are provided to analyze the effects of power law exponent, boundary conditions, temperature dependency, side to radius ratio, and side to thickness ratio.     

Nonlinear dynamic response and vibration of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shear deformable plates with temperature-dependent material properties and surrounded on elastic foundations

Based on Reddy’s third-order shear deformation plate theory, the nonlinear dynamic response and vibration of imperfect functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates on elastic foundations subjected to dynamic loads and temperature are presented. The plates are reinforced by single-walled carbon nanotubes which vary according to the linear functions of the plate thickness. The plate’s effective material properties are assumed to depend on temperature and estimated through the rule of mixture. By applying the Airy stress function, Galerkin method and fourth-order Runge–Kutta method, nonlinear dynamic response and natural frequency for imperfect FG-CNTRC plates are determined. In numerical results, the influences of geometrical parameters, elastic foundations, initial imperfection, dynamic loads, temperature increment, and nanotube volume fraction on the nonlinear vibration of FG-CNTRC plates are investigated. The obtained results are validated by comparing with those of other authors.     

Vibration and Critical Speed of Orthogonally Stiffened Rotating FG Cylindrical Shell Under Thermo-Mechanical Loads Using Differential Quadrature Method

In this article, an approximate solution using differential quadrature method is presented to investigate the effects of thermo-mechanical loads and stiffeners on the natural frequency and critical speed of stiffened rotating functionally graded cylindrical shells. Transverse shear deformation and rotary inertia, based on first-order shear deformation shell theory (FSDT), are taken into consideration. The equations of motion are derived by the Hamilton's principle while the stiffeners are treated as discrete elements. Material properties are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fraction of the constituents. The temperature field is assumed to be varied in the thickness direction. The equations of motion as well as the boundary condition equations are transformed into a set of algebraic equations applying the DQM. The results obtained include the relationship between frequency characteristics of different power-law index, rotating velocities, thermal loading and amplitude of axial load. To validate the present analysis, the comparison is made with a number of particular cases in literature. Excellent agreement is observed and a new range of results are presented for stiffened rotating FG cylindrical shell under thermo-mechanical loads which can be used as a benchmark to approximate solutions.     

Thermal Instability of Functionally Graded Shallow Spherical Shell

In this paper, thermal instability of shallow spherical shells made of functionally graded material (FGM) is considered. The governing equations for a thin spherical shell based on the Donnell–Mushtari–Vlasov theory are obtained. The equations are derived using the Sanders simplified kinematic relations and variational method. It is assumed that the mechanical properties vary linearly through the shell thickness. The constituent material of the functionally graded shell is assumed to be a mixture of ceramic and metal. Analytical solutions are obtained for three types of thermal loading including Uniform Temperature Rise (UTR), Linear Radial Temperature (LRT), and Nonlinear Radial Temperature (NRT). The results are validated with the known data in the literature.

     

ASSESSMENT OF SHELL THEORIES FOR HYBRID PIEZOELECTRIC CYLINDRICAL SHELL UNDER THERMOELECTRIC LOAD

This study presents an assessment of classical lamination shell theory and first-order shear deformation theory for a simply supported finite circular cylindrical hybrid shell with a cross-ply laminate as an elastic substrate under thermoelectric static load. Navier-type solutions for these shell theories are obtained and used in three-dimensional (3D)equilibrium equations and transverse strain-displacement relations to obtain transverse stress components and an improved value of deflection. These solutions are assessed by comparison with the 3D solution.     

Thermomechanical nonlinear analysis of axially compressed carbon nanotube-reinforced composite cylindrical panels resting on elastic foundations with tangentially restrained edges

Buckling, postbuckling, and nonlinear responses of composite cylindrical panels reinforced by single-walled carbon nanotubes (CNTs), supported by an elastic foundation, exposed to elevated temperature and axially compressed by uniform load are investigated in this article. Distribution of CNTs is uniform or graded in the thickness direction and the effective properties of CNT-reinforced composite are assumed to be temperature dependent, and are estimated by extended rule of mixture through a micromechanical model. Governing equations are established based on thin shell theory taking von Kármán–Donnell nonlinearity, initial geometrical imperfection, Pasternak-type elastic foundation and tangential elastic constraints of boundary edges into consideration. Approximate solutions of deflection and stress functions are assumed to satisfy simply supported boundary conditions, and Galerkin method is applied to derive explicit expressions of load–deflection relation from which critical buckling loads can be obtained. Unlike works in the literature, the present study accounts for elasticity of tangential restraint of two unloaded straight edges in model of cylindrical panel. The study also gives conditions for which bifurcation type buckling response can occur and novel findings in numerical examples.     

Buckling of Functionally Graded Cylindrical Shells Under Combined Thermal and Compressive Loads

This article focuses on analytical solutions for bifurcation buckling of FGM cylindrical shells under thermal and compressive loads. A new solution methodology is established based on Hamilton's principle. The fundamental problem is subsequently transformed into the solutions of symplectic eigenvalues and eigenvectors, respectively. Then, by applying a unidirectional Galerkin method, imperfection sensitivity of an imperfect FGM cylindrical shell is discussed in detail. The solutions reveal that boundary conditions, volume fraction exponent, FGM properties, and temperature rise distribution significantly influence the buckling behavior. Critical stresses are reduced greatly due to the existence of initial geometric imperfections.     

Thermal Buckling of Functionally Graded Plates Resting On Elastic Foundations Using the Trigonometric Theory

In this article, thermal buckling analysis of functionally graded material (FGM) plates resting on two-parameter Pasternak's foundations is investigated. Equilibrium and stability equations of FGM plates are derived based on the trigonometric shear deformation plate theory and includes the plate foundation interaction and thermal effects. The material properties vary according to a power law form through the thickness coordinate. The governing equations are solved analytically for a plate with simply supported boundary conditions and subjected to uniform temperature rise and gradient through the thickness. Resulting equations are employed to obtain the closed-form solution for the critical buckling load for each loading case. The influences of the plate aspect ratio, side-to-thickness ratio, gradient index, and elastic foundation stiffnesses on the buckling temperature difference are discussed.     

Nonlinear static behaviour of shallow spherical shell on a Kerr foundation

A large deflection analysis of a shallow spherical shell on a Kerr foundation is carried out for a uniformly distributed load and for different edge conditions, using the Sinharay and Banerjee approach. Detailed parametric studies involving variations in geometric, material and foundation properties are made; load-deflection curves for shells on a Kerr foundation are compared with those for shells on an equivalent Pasternak foundation. This study shows that the deflection reduces if the curvature, the Poisson's ratio or the foundation parameters of the shear and lower spring layer are increased. Shells with immovable edges are found to be more prone and vulnerable to changes in various parameters than those with movable edges.