Bending deflection and stress analyses in a thin functionally graded material skew plate with different boundary conditions on the Winkler–Pasternak elastic foundation and under combined in-plane and uniform lateral loads using the extended Kantorovich method

In this study, bending deflection and stress analyses have been conducted for a thin skew plate made of functionally graded material (FGM) with different boundary conditions on the Winkler–Pasternak elastic foundation and under combined loads including uniform transverse load, normal and shear in-plane forces, and planar body forces. The Cartesian partial differential equation governing the bending deflection of the skew plate has been converted into a partial differential equation in oblique coordinates using the conversion relations. Then, by employing the variational principle and residual weighted Galerkin method and using the Extended Kantorovich Method (EKM), the equation has been converted to a set of linear differential equations in terms of two functions in the longitudinal and transverse directions of the oblique plate, and afterward, the equation has been solved using the iterative solution method. Different boundary conditions in a combined form of simply and clamped supports have been investigated and their effects on bending deflection and generated in-plane normal and shear stresses are discussed.

The correct derivation of the buckling equations of the shear-deformable FGM plates for the extended Kantorovich method

This article presents the derivation of the elastic buckling equations and boundary conditions of shear-deformable plates in the frame of the extended Kantorovich method (EKM). Surveying the literature shows that those stability equations are often obtained using a wrong derivation by confusing them with the linear equilibrium condition. This work aims at providing the correct derivation that is built on the stability of the equilibrium condition. Buckling equations are derived for three different plate theories, namely, the first-order shear deformation plate theory (FSDT), the refined-FSDT, and the refined plate theory (RPT). This article is the first to implement the EKM based on a refined theory. Also, it is the first time to implement the refined-FSDT in buckling analysis. For the generic FGM plates, buckling equations derived based on the FSDT and refined-FSDT are both found to be simple and contain only the lateral displacements/rotations variations. On the other hand, those of the RPT, have coupled lateral and in-plane displacement variations, even if the physical neutral plate is taken as the reference plane. The considered plate is rectangular and under general in-plane loads. The properties are of the plate continuously varying through its thickness which is assumed to change smoothly with a separable function in the two in-plane dimensions. The von Kármán nonlinearity is considered. The stability equations are derived according to the Trefftz criterion, using the variational calculus. The solution methods of the obtained equations are out of the scope of this article.