General nonlinear plate theory applied to a circular plate with large deflections

The general nonlinear 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. However, the equations do not reduce to the Reissner equations for large deflections.

The solution to the problem of a clamped circular plate loaded with a concentrated load on a central rigid inclusion was obtained and agreed with the solution of von Karman'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 for this particular problem. 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radius, the boundary layer approaches the order of magnitude of the thickness and thus a first approximation theory is no longer valid.

Two membrane problems are also solved. The first is that of a circular membrane deformed by a load which acts normal to the plane of a central rigid inclusion. A closed form solution is obtained for this problem when Poisson's ratio is equal to 1/3. An approximate solution is obtained for any value of Poisson's ratio for the case where the deflections are very large. The second problem is the same as the first with the addition of a small torque about a normal to the rigid inclusion. An approximate solution is obtained to this problem.