Cantilevered Plate Rayleigh-Ritz Trial Function Selection for von Kármán's Plate Equations

Abstract

The accuracy and convergence characteristics of the classical Rayleigh-Ritz solution of the nonlinear von Karman plate equations are studied with respect to the types of cantilever plate in-plane trial functions used in the solution. The static deflection of the cantilever plate is computed for an applied static gravity loading. Four different in-plane trial function types are studied. In each of these cases the same out-of-plane trial functions are used. It is found that in two of the four cases good convergence and accuracy are achieved when compared to the solution from a nonlinear finite element model. The degree of satisfaction of the problem natural boundary conditions is also examined, and it is shown that for the two cases that show inadequate convergence characteristics this satisfaction is poor. It is noted in particular that for these two cases at points on the problems' free boundaries, the in-plane trial functions satisfy, either exactly or approximately, the linear in-plane natural boundary conditions. At these points the addition of inplane degrees of freedom to the solution will not contribute to the satisfaction of the nonlinear natural boundary condition. Thus for these two cases the convergence is poor as more in-plane trial functions are added to the solution. The change in the statically loaded plate natural frequencies are also computed. Similar to the static deflection results, convergence to the nonlinear finite element solution is slow when in-plane trial functions are used that approximately satisfy the linear in-plane boundary conditions.