The buckling of heated functionally graded material (FGM) annular plates on an elastic foundation is studied analytically. A conventional Pasternak-type elastic foundation is assumed to be in contact ...with plate during deformation, which acts in both compression and tension. The equilibrium equations of an annular-shaped plate are obtained based on the classical plate theory. Each thermo-mechanical property of the plate is assumed to be graded across the thickness direction of plate based on the power law form, while Poisson’s ratio is kept constant. Among all combinations of free, simply-supported, and clamped boundary conditions, existence of bifurcation buckling for various edge supports is examined and stability equations are obtained by means of the adjacent equilibrium criterion. An exact analytical solution is presented to calculate the thermal buckling load by obtaining the eigenvalues of the stability equation. Three types of thermal loading, namely; uniform temperature rise, transversely linear temperature distribution and heat conduction across the thickness type are studied. Effects of thickness to outer radii, inner to outer radii, power law index, elastic foundation coefficient, and thermal loading type on critical buckling temperature of FG plates are presented.
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.
In this study buckling analysis of a functionally graded conical shell integrated with piezoelectric layers that is subjected to combined action of thermal and electrical loads is presented. The ...material properties of functionally graded conical shells are assumed to vary continuously through the thickness direction based on a power law form. The governing equations, including the equilibrium and stability equations, are obtained based on the classical shell theory and the Sanders nonlinear kinematics relations. The case of uniform temperature distribution through the shell domain is considered. The prebuckling forces are obtained considering the membrane solutions of linear equilibrium equations. Minimum potential energy criterion is employed to establish the stability equations. The single-mode Galerkin method is used to obtain the critical buckling temperature difference. The results are compared with the known data in the open literature. Finally, some numerical results are presented to study the effects of applied actuator voltage, shell geometry, and power law index of FGM on thermal buckling behavior of the conical shell.
This study presents the buckling analysis of radially loaded solid circular plate made of porous material. Properties of the porous plate, where pores are assumed to be saturated with fluid, vary ...across its thickness. The boundary condition of the plate is assumed to be clamped and the plate is assumed to be geometrically perfect. The higher order shear deformation plate theory (HSDT) is employed to derive the governing equations. The equilibrium and stability equations, derived through the variational formulation and based on the Sanders non-linear strain–displacement relation, are used to determine the prebuckling forces and critical buckling loads. The results are compared with the buckling loads of circular plates made of porous material and reported in the literature based on the classical plate theory (CPT) and the first order shear deformation plate theory (FSDT).
•We analyze the buckling of functionally graded circular plates made of porous material on higher order shear deformation theory.•The porous plate is assumed of the form that pores are saturated with fluid.•And the pores distribution of the plate is variable in the thickness direction and the plate is investigated in three situations.•Equilibrium and stability equations of a porous circular plate under radially compressive load are derived.•The effects of porous plate, thickness, pores distribution and variation of porosity on the critical mechanical load are investigated.
Considering the uncoupled thermoelasticity assumptions, a thermally induced vibration analysis for functionally graded material (FGM) conical shells is performed in this research. Thermo-mechanical ...properties of the conical shell are assumed to be temperature and position dependent. The conical shell is under rapid heating with various cases of thermal loads on the outer surface, whereas the opposite surface is kept at reference temperature or thermally insulated. Since the ratio of thickness to radius is much smaller than one, the transient heat conduction equation, for simplicity, may be established and solved for one-dimensional condition. Assuming temperature-dependent material properties, the heat conduction equation is nonlinear and should be solved using a numerical method. A hybrid generalized differential quadrature (GDQ) and Crank–Nicolson method is used to obtain the temperature distribution in thickness direction, respectively. Based on the first-order shear deformation theory and geometrically nonlinear assumptions, the equations of motion are obtained applying the Hamilton principle. Discretization of the equations of motion in the space domain and boundary conditions is performed by applying the GDQ method, and then, the system of highly nonlinear coupled ordinary differential equations is solved by the iterative Newmark time-marching scheme and well-known Newton–Raphson method. Since the thermally induced vibration of the conical shells is not reported in the literature, the results are compared with the case of a circular plate. Also, studies of the FGM conical shells for various types of boundary conditions, functionally graded patterns, and thermal loads are provided. The effects of temperature dependency, geometrical nonlinearity, semi-vertex angle, shell length, and shell thickness upon the deflections of the conical shells are investigated.
The present research deals with the thermoelastic response of a thick sphere based on the Lord–Shulman theory of generalized thermoelasticity. Unlike the other available works in which energy ...equation is linearized, the assumption of ignorance of temperature change in comparison to the reference temperature is not established in this research resulting in a nonlinear energy equation. Such nonlinearity is called thermally nonlinear. The one-dimensional radial equation of motion and energy equation are established for an isotropic homogeneous sphere. The resulting equations are discreted by means of the generalized differential quadrature in radial direction and traced in time by means of the Newmark time marching scheme. Numerical results are provided to demonstrate the discrepancies between the thermally linear and nonlinear results. As the numerical results reveal, thermally linear theory fails for precise analysis of structures under thermal loads especially at high temperature shocks, large coupling coefficient, and large relaxation time.
•Lord–Shulman theory is used to avoid the infinite speed of thermal wave propagation.•A thermally nonlinear analysis is performed where the energy equation is kept in its complete form.•Generalized differential quadrature is implemented to discrete the motion and energy equations.•Linearization of energy equation fails for severe thermal shocks, high relaxation time and high coupling parameter.
This study deals with the response of an FGM annular plate under lateral thermal shock load. The equations of motion are obtained using the first order shear deformation plate theory. The governing ...equations are solved using the Laplace transformation and Galerkin finite element method. Finally, numerical inversion of the Laplace transform is carried out to obtain the results in real time domain. It is shown that coupling coefficient has a damping effect on the radial force resultant and deflection.
In the present work, a nonlinear thermo-electro-mechanical response of functionally graded piezoelectric material (FGPM) actuators is investigated. The theoretical formulation is based on the ...Timoshenko beam theory with the von Kármán nonlinearity (in the form of midplane stretching), and a microstructural length scale is incorporated by means of the modified couple stress theory. A power-law distribution of thermal, electrical, and mechanical properties through beam thickness (or height) is assumed. The governing equations are derived using the principle of virtual displacements. A displacement finite element model of the theory is developed, and the resulting system of nonlinear algebraic equations is solved with the help of Newton's iteration method. Numerical results are presented for transverse deflection as a function of load parameters and out-of-plane boundary conditions. The parametric effects of microstructural length scale parameter, power-law index of the material distribution across the thickness, boundary conditions, beam geometry, and applied actuator voltage on the beam response are investigated through various numerical examples. The results reveal the existence of bifurcation (or critical states) for certain types of in-plane loads. For other load types, including out-of-plane loads, the beam undergoes a unique and stable deflection path that does not contain any critical point.
•Accounts for shear deformation with geometric nonlinearity and size effects.•Buckling analysis with identification of critical states is carried out.•Includes the effects of microstructure dependency, and material distribution.•Includes the effects of boundary conditions, beam geometry, and applied voltage.
AbstractThis study presents the buckling analysis of a radially loaded, solid, circular plate made of porous material. Properties of the plate vary across the thickness. The edge of the plate is ...either simply supported or clamped and the plate is assumed to be geometrically perfect. The geometrical nonlinearities are considered in the Love-Kirchhoff hypothesis sense. The equilibrium and stability equations, derived through the variational formulation, are used to determine the prebuckling forces and critical buckling loads. The equations are based on the Sanders nonlinear strain-displacement relation. The porous plate is assumed to be of the form where pores are saturated with fluid. The results obtained for porous plates are compared with the homogeneous and porous/nonlinear, symmetric distribution, circular plates.
The present research considers the free vibration characteristics of a joined shell system that consists of three segments. The joined shell system contains two conical shells at the ends and a ...cylindrical shell at the middle. All shell elements are made from isotropic homogeneous material. The shell elements are unified in thickness. With the aid of the first-order shear deformation shell theory and the Donnell type of kinematic assumptions, the equations of motion of a conical shell and the associated boundary conditions are obtained. These equations are valid for each segment. The obtained equations are then discreted using the generalised differential quadratures (GDQ) method. Applying the intersection continuity conditions for displacements, rotations, forces, and moments between two adjacent shells, and also boundary conditions at the ends of the joined shell system, a set of homogeneous equations is obtained, which governs the free vibration motion of the joined shell. Comparisons are made with the available data in the open literature for the case of thin conical–cylindrical–conical shells with special types of geometry or boundary conditions. Afterwards, numerical results are provided for moderately thick shells with different geometrical and boundary conditions.
