Analysis of Plate Examples by Difference Methods and the Superposition Principle

In this paper the author applies the membrane analogs of H. Marcus to some elementary cases of thin homogeneous isotropic square plates having central-point loads and various boundary conditions. The analogy is made possible by two theorems: (a) The deflection of a membrane loaded with loads proportional to those on a given plate may be considered as the sum of the principal moments of the actual plate. (b) A second membrane may be loaded with elastic weights proportional to these moment sums and, subject to appropriate boundary conditions, the deflections of the latter membrane will be proportional to the deflections of the actual plate under the given loading system. The principle of superposition of deflection surfaces or equivalent stress systems is utilized in this paper both by difference and differential methods. The problems treated are (1) a square plate with pinned or simply supported edges, (2) two opposite edges pinned and the other two free, (3) two opposite edges pinned and the other two clamped, (4) all four edges clamped, (5) all four edges free with only corner post supports. The correct critical stress at the center of the lower surface of the plate was obtained from special thick-plate theory for a particular thickness-to-span ratio. The effect of this critical stress on the whole plate action is depicted for various boundary conditions.