Abstract
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 of the plate are continuously varying through its thickness which is assumed to change smoothly with a separable function in the two in-plane directions. 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, however, a brief on the solution strategy is presented.
Highlights
The extended Kantorovich method (EKM) had evolved from the so called classical Kantorovich method that was introduced in [11] as a solution method to the partial differential equations (PDE) by reducing the d-dimensional PDE that governs the problem into d one dimensional (1D) ordinary differential equations (ODE)
The buckling equations and the boundary conditions of rectangular plates are derived in the frame of the EKM
The formulations are based on the first-order shear deformation theory (FSDT), the refined-FSDT, and the refined plate theory (RPT)
Summary
The extended Kantorovich method (EKM) had evolved from the so called classical Kantorovich method that was introduced in [11] as a solution method to the partial differential equations (PDE) by reducing the d-dimensional PDE that governs the problem into d one dimensional (1D) ordinary differential equations (ODE) Obtaining those ODE’s starts by approximating the d-dimensional solution function Ψ The limitation of the classical Kantorovich method is that its accuracy depends on the initially guessed functions To overcome this limitation, [12] proposed the extended Kantorovich method (EKM), in which the obtained ODE’s are solved iteratively until a satisfying convergence is achieved. By using n terms in the approximate solution, the d-dimensional PDE is reduced to d sets, each contains n ODE’s that have to be solved simultaneously
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