Ph.D Thesis
Ph.D Student Shufrin Igor A Semi-Analytical Solution for Linear and Nonliner Analysis of Plates Department of Civil and Environmental Engineering Professor Moshe Eisenberger Professor Oded Rabinovitch

Abstract

In this study, a general semi-analytical solution for the linear and nonlinear analysis of plates with arbitrary boundary conditions is developed. The approach is based on the variational principle of virtual work and the extended Kantorovich method (EKM). The elastic thin plate theory with the geometrically nonlinear von Kármán strains is adopted for the analysis. In order to overcome the limitation of the classical single-term formulation of the EKM, a general multi-term formulation is derived. The applicability, accuracy and convergence of the multi-term extended Kantorovich method (MTEKM) are studied through linear analyses first. Bending, free vibration, in-plane stress, and stability analyses of rectangular plates are discussed. Emphasis is placed on combinations of mechanical properties, boundary, and loading conditions that cannot be analyzed using the classical single-term EKM.

Next, a general semi-analytical approach for the geometrically nonlinear analysis of plates is developed. The efficiency and convergence of the MTEKM for the nonlinear large deflection analysis of rectangular laminated composite plates with general combinations of boundary and loading conditions are examined through a comparison with other semi-analytical methods and with finite element analyses.

Special attention is drawn to the geometrically nonlinear analysis of rectangular plates subjected to destabilizing in-plane loads. In order to apply the approach developed here, parameter continuation and the pseudo-arc-length continuation procedures for the solution of a system of nonlinear partial differential equations are developed. These procedures are generalized and augmented for the application to PDE and for the solution in the space of unknown functions of two variables. 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 derived continuation procedures is demonstrated through the solution of the Bratu-Gelfand benchmark problem. The ability of the method to capture the mode-jumping phenomenon in plates subjected to destabilizing in-plane loads is demonstrated.

Finally, the semi-analytical MTEKM for the nonlinear large deflections analysis of skew and trapezoidal plates is developed. The trapezoidal geometry is mapped into a rectangular computational domain. The applicability and convergence of the method to the large deflection analysis of skew and trapezoidal plates are studied numerically. Emphasis is placed on the combinations of geometry, loading, and boundary conditions that are beyond the applicability of other semi-analytical methods.