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In this study, a simply and efficient closed form solution for bending and stress analysis of functionally graded circular and annular plates with elastic boundary conditions is presented based on the first order shear deformation theory (FSDT). By using the presented solution procedure, functionally graded plates subjected to arbitrary non-uniformly distributed normal and shear loads may be analuzed and all of stresses components may be exactly achieved. Shear loads may be imposed on the top and bottom surfaces of plate. By using the constitutive equations based on the first-order shear deformation theory, the transverse shear stress components cannot be obtained correctly and constant through-the-thickness distributions will be extracted. So, to achieve the transverse normal and shear stresses components in the proposed solution procedure, the three dimensional theory of elasticity is applied. To establish the accuracy and efficiency of the proposed approach, the obtained results are compared with other available published results and results of the three-dimensional theory of elasticity extracted from the ABAQUS software based on the finite element method (as the most exact method). Comparisons show that the obtained results are very accurate, while it is computationally quite more economic than the three-dimensional elasticity approach. Also, transverse normal and shear stresses boundary conditions on the top and bottom surfaces of the plate are exactly satisfied. Even for a complicated loading, when the non-uniform normal and shear loads are imposed simultaneously on the top and bottom surfaces of plate and transverse stresses boundary conditions on these surfaces are non-zero.

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In this study, an improved third order shear deformation theory is used to analyze the thermoelastic buckling of a functionally graded rectangular plate. The plate is assumed to be under two types of thermal loading, namely, uniform temperature rise across the thickness and linear temperature change across the thickness of the plate. Moreover, the material properties of the functionally graded plate vary linearly through the thickness and simply supported are considered for all edges of the plate. First, the nonlinear strain-displacement relations are considered based on improved third order theory and then the equilibrium and stability equations are derived. In continue, displacements and the pre-buckling forces are calculated using the equilibrium equations. The temperature difference relation of buckling is obtained by solving the stability equations. To obtain the critical temperature difference, the recent relation is minimized with respect to the number of half wave parameters. Resulting equations are compared with the literature. The results show that, the values of temperature difference buckling obtained based on improved third order shear deformation theory, are lower compared with the classical plate theory, first and third order shear deformation theories. Moreover, the value of critical temperature difference under linear temperature change is bigger compared with the uniform temperature rise across the thickness, and the difference between the two values will be bigger with increasing the thickness of the plate.

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The temperature-dependent nonlinear mechanical behaviors of functionally graded rectangular plates in the thickness direction resting on Winkler–Pasternak elastic foundation are investigated using the three-dimensional theory of elasticity. The material properties are temperature-dependent and varied in the thickness direction based on a power-law. Considering the nonlinear Green-Lagrange strain relation, the geometric nonlinearity is taken into account. After obtaining the potential strain, kinetic energies, taking into account the effects of the temperature and the elastic foundation, the Hamilton’s principle is used to derive the nonlinear three-dimensional governing equations and corresponding boundary conditions. To solve the nonlinear free vibration problem, first, the generalized differential quadrature (GDQ) method is used to discretize the nonlinear coupled governing equations in the space domain. Then, the obtained equations are converted to the time-dependent ordinary differential equations using the numerical-based Galerkin scheme and the time periodic discretization (TPD) are used to discretize them in the time domain. Finally, the arc-length method is employed to find the frequency-response of system. Also, to solve the nonlinear bending problem, by neglecting the effect of inertia and using the arc length algorithm, the maximum deflection versus the applied load is obtained. The effects of different parameters such as length-to-thickness ratio, Winkler–Pasternak elastic foundation coefficients, uniform and linear temperature rises and volume fraction index on the frequency response and maximum deflection of functionally graded plates with various edge conditions are studied.