NACA-TN-1425

Abstract

The present report presents a theoretical analysis of an initially flat, rectangular plate with large deflections under either normal pressure or combined normal pressure and side thrust. As small deflections of a flat plate are governed by a single linear equation, large deflections introduce nonlinear terms into the conditions of equilibrium and are governed by 2 fourth-order, second-degree, partial differential equations. These 30-called Von K&#225;rm&#225;n equations are studied in the present report by use of the finite-difference approximations. The difference equations are solved by two methods, namely, the method of successive approximations and the relaxation method. Neither of these methods is new, but their application to nonlinear problems requires new techniques. <br>The problem of a uniformly loaded square plate with boundary conditions which approximate the riveted sheet-stringer panels is solved by the method of successive approximations. The theoretical center deflections show good agreement with the recent experimental results obtained at the California Institute of Technology when the deflections are of the order of the plate thickness. This agreement perhaps suggests the range in which these Von K&#225;rm&#225;n equations are to be applied. <br>Other problems of thin plates with large deflections are discussed from the point of view of an aeronautical engineer. The boundary conditions which approximate the various cases are formulated, and the methods for solving these problems are outlined. <br>Since the .method presented in the present report is general, It may be applied to solve bending and combined bending and buckling problems with practically any, boundary conditions, and the results may be obtained to any degree of accuracy required. Furthermore, the same method may be applied to solve the membrane theory of the plate which applies when the deflection is very large In comparison with the thlckness of the plate.