Linear and nonlinear deflection analysis of thick rectangular plates using finite differences.

Abstract

Variational methods are widely used for the solution of complex differential equations in mechanics for which exact solutions are not possible. The finite difference method, although well known as an efficient numerical method was applied in the past only for the solution of linear and nonlinear thin plates. In the present study, the suitability of the method for the solution of nonlinear deflection of thick plates is studied for the first time. While there is major differences between small deflection and large deflection plate theories, the former can be treated as a particular case of the latter, when the centre deflection of the plate is less than or equal to 0.2-0.25 of the thickness of the plate. The finite difference method as applied here is a modified finite difference approach to the ordinary finite difference method generally used for the solution of thin plate problems. In this thesis thin plates are treated as a particular case of the corresponding thick plate when the boundary conditions of the plates are taken into account. The method is first applied to investigate the deflection behaviour of square clamped and simply supported square isotropic thick plates. After the validity of the method is established, it is then extended to the solution of rectangular thick plates of various aspect ratios and thicknesses. Generally, beginning with the use of a limited number of mesh sizes for a given plate aspect ratio and boundary conditions, a general solution of the problem including the investigation of accuracy and convergence was extended to rectangular thick plates by providing more detailed functions satisfying the rectangular mesh sizes generated automatically by the programme. Whenever possible results of the present method are compared with the existing solutions in the technical literature obtained by much more laborious methods and close agreements are found. Significant amounts of results presented herein are not currently available in the technical literature for various plate aspect ratios and Poisson's ratios. The submatrices involved in the formation of the finite difference equations from the governing differential equations forming the general system are generated directly by the computer programme. The subroutine SOLINV from the second directed method as developed and illustrated in Chapter V takes care of the inversion of the general matrix. The subroutine developed by the author has been proven to be more efficient than the former methods known for the computation of linear simultaneous equations [61]. Simplicity in formulation and quick convergence are the obvious advantages of the finite difference formulation developed here for small and large deflection analysis of thick plate in comparison with other numerical methods requiring extensive computer facilities.