Buckling of plates by the multi term extended Kantorovich method

The linear and nonlinear stability analysis of rectangular plates is developed using the thin plate theory with the nonlinear von Kármán strains. The multi-term extended Kantorovich method is employed to transform the coupled nonlinear set of equilibrium equations into a set of nonlinear equations. For linear analysis the solution is obtained using the exact element method, and for the nonlinear case it is also combined with solution methodologies which are required for tracking a highly nonlinear equilibrium path, namely: parameter continuation and arc-length continuation procedures. These procedures, which are commonly used for the solution of discretized structural systems governed by nonlinear algebraic equations, are augmented and generalized for the direct application to the PDE. The boundary value problem that results from the arc-length continuation scheme and consists of coupled differential, integral, and algebraic equations is re-formulated in a form that allows the use of standard numerical BVP solvers. The applicability of the proposed approach to the solutions of many examples is presented for rectangular plates with various boundary conditions and other complicating effects