Application of Variation Method in Three Dimensional Stability Analysis of Rectangular Plate Using Various Exact Shape Functions

Abstract

In this paper, a polynomial and trigonometric shape function are developed for the three-dimensional (3-D) stability analysis of a thick rectangular plate. This study has evaluated the effect of aspect ratio of the critical buckling load of a plate that is clamped on the first edge, free at the third edge, with the second and fourth edges simply supported respectively (CSFS) using a variational method. An expression of potential energy of thick plate was formulated using 3-D elastic principles thereafter, a compatibility equation of 3-D plate was derived through energy equation transformation to get the relations between the rotations and deflection. The solution of compatibility equations yields the exact plates shape function which is derived in terms of trigonometric and polynomial displacement and rotations. Similarly, by minimizing the energy equation with respect to the deflection, the direct governing equation was formulated. The solution of governing equation yields the deflection coefficient of the plate. By minimizing the potential energy equation with respect to deflection coefficient after the action deflection and rotations equation were substituted into it, a more realistic formula for calculation of the critical buckling load is established. The proposed method unlike the refined plate theory (RPT), considered all the six stress elements in the analysis. The result showed that the critical buckling loads from the present study using polynomial are slightly higher than those obtained using trigonometric theories signifying the more exactness of the latter. The result of the present study using the established 3-D model for both functions is satisfactory and closer to exact solution compared to the two- dimensional (2-D) RPT. The overall average percentage differences between the two functions recorded are 6.4%. This shows that at about 94% both approaches are the same and can be applied with confidence in the stability analysis of any type of plate with the boundary condition.