Non Linear Analysis of Isotropic Rectangular Thin Plates Using Ritz Method

ABSTRACT  

This research work presents Nonlinear Analysis of Isotropic Thin Rectangular Plates using Energy Principle (Ritz method). Isotropic thin Rectangular plate having different twelve boundary conditions were analyzed and these boundary conditions were formed by combination of three major supports – Clamp, C; Simply supported, S; and Free, F. With this combination, the twelve boundary conditions analyzed include: SSSS, CCCC, CSCS, CSSS, CCSS, CCCS, CCFC, SSFS, CCFS, SCFC, CSFS, and SCFS. General expressions for displacement and stress functions for large deflection of isotropic thin rectangular plate under uniformly distributed transverse loading were obtained by direct integration of Von karman’s non-linear governing differential compatibility and equilibrium equations. Polynomial function instead of trigonometry function as was with previous researchers was used on the decoupled Von Karman’s equations to obtain particular stress and displacement functions respectively. Non-linear total potential Energy was formulated using Von Karman equilibrium equation and Ritz method was deployed in this formulation. This equation was fully converted to potential energy by multiplying all the terms in it with displacement, wand the formed total potential energy, π consists of potential energy of internal forces and potential energy of external forces. This formulated total potential energy π, could give an accurate approximation of displacement field if the parameters were properly chosen.However, we assumed deflection, w to be ∆H1, and stress function, ɸ to be ∆ 2H2 and substituted into the formulated potential energy. H1and H2 are profiles of the deflection and stress function respectively, and ∆ is deflection coefficient factor of the plate. Potential energy formulated contains deflectioncoefficient factor to the power of four.This potential energy was minimized by differentiating it partially with respect to coefficient factor reducing to cubic form. Particular stress and displacement functions were substituted into the minimized nonlinear total potential energy which resulted to general amplitude equation of the form K1∆ 3 +K2∆+K3 for each of the twelve boundary conditions considered. Where K1, K2 and K3are coefficients of amplitude equation. K1 and K2are dependent on aspect ratio. Aspect ratios considered ranged from 0.1 to 0.5 with an increment of 0.1. Newton-Raphson method was used to evaluate the deflection coefficient factor. Results obtained were compared with the available results from the previous researchers. The comparison made are only for SSSS, CCCC and CCCS plates. It was so because the remaining boundary conditions considered in this work have not been researched upon by previous researchers. From results obtained, the average percentage differences recorded for SSSS, CCCC, and CCCS plates for the present and previous studies are4.01978%, 3.7646%, and 5.02%respectively. The percentage differences for the three plates compared are within acceptable limit of 0.05 or 5% level of significance in statistics. From the comparison made, it was obvious that an excellent agreement was observed in all cases thus indicating applicability and validity of the polynomial function for solving exact plate bending problems.In addition, nonlinear analysis of the twelve boundary conditions was carried out under three different loads. These analyses revealed the behavior and dynamics of these plates. It was discovered from the results that clamped edges possess more stiffness than the simply supported and the free edges. This is evident in the ease at which the free edge stretches. Expressions for bending moment, shear forces, inplane forces and stress distributions were established as well. Computer program was developed and this program is capable of determining deflection and stresses at any point of the plate against the usual method of evaluating deflection at the center.