General non-linear plate theory applied to a circular plate with large deflections

Abstract

The general non-linear first approximation thin plate tensor equations in undeformed coordinates valid for large strains, rotations and displacements are developed based on the single assumption of plane stress. These equations are then reduced to the exact tensor and physical component equations for symmetrical circular plates. An order of magnitude analysis is performed on the resulting equations which shows that they reduce to the classical linear equations for very small deflections and to the von Karman equations for moderate deflections. The solution to the problem of a clamped circular plate loaded with a concentrated load on a central rigid inclusion is obtained and agrees with the solution of von Kármán's equations for moderate deflections. Perhaps the most important result is that of finding the order of magnitude of the limiting value of deflection that would be allowed under the assumption of plane stress. It is shown that when the deflection approaches the order of magnitude of the radius, the boundary layer approaches the order of magnitude of the thickness and thus a first approximation theory is no longer valid.