Abstract:
This research work presents the analysis of rectangular plate on Winkler’s foundation using second degree characteristic orthogonal polynomials. The plates were of three different edge conditions: Thin Rectangular Plate pinned on all edges (SSSS), Thin Rectangular Plate clamped on all edges (CCCC) and Thin Rectangular Plate pinned on two opposite edges and clamped on the other two edges (CSCS). The exact displacement and bending moment functions of these plates under uniformly distributed in–plane loading on their longitudinal edges on Winkler foundation were obtained by applying work principle approach and total minimum energy method. These methods were applied on the governing differential equation of plates so as to obtain the deflection coefficient Wuv equations. These coefficients were obtained by finding the second degree of freedom functional equation and substituting them on work done expression for plates putting into consideration the aspect ratio of the rectangular plate. The expression gotten were later rearranged in matrix form to give the stiffness matrix of the plates which were later substituted back to the defection coefficient equation, Wuv The bending moments were then obtained by substituting the shape functions in the moment equation expressed in non-dimensional parameters and aspect ratio. The values of midspan deflection gotten for SSSS plate as aspect ratio varies from 1.0 to 2.0 are (0.000062590, 0.000062748, 0.000062859, 0.000062939, 0.000063000, 0.000063082, 0.000063111, 0.000063135,and 0.000063154)m. The difference from Timoshenko and Woinowsky values reflects that second degree characteristic orthogonal polynomials produces better results when used to analyse higher degree polynomials. This also featured on results for CCCC plate-(0.000065330, 0.000065680, 0.000065923, 0.000066098, 0.000066227, 0.000066325, 0.000066401, 0.000066460, 0.000066508, and 0.000066546) m and CSCS plate-(0.0011881, 0.0011898, 0.0011910, 0.0011917, 0.0011923, 0.0011927, 0.0011930, 0.0011932, 0.0011934, and 0.0011935) m. In conclusion higher degrees of freedom Characteristic orthogonal polynomials shape functions for rectangular plates are satisfactory in approximating the deformed shape of thin rectangular plates of various boundary conditions. The results from the study compares very well with those of the literature. Though they are upper bound, they are adequate to be used in design as they are safer and straight forward.
