While solutions to the nonlinear von Karman equations for large deflections of thin plates have been presented for annular plates under certain axisymmetric/loading conditions, little work has been done with unsymmetrical large deflections. In this investigation a systematic mathematical iteration technique is used to obtain a solution to the von Karman equations for an annulus fixed at the outer edge and which has a central rigid plug that is rotated about its diameter out of the plane of the plate.

The linear, small-deflection solution to this problem presented by H. Reissner is used as the first approximation for large deflections. By using Reissner's solution for the lateral displacement to evaluate the nonlinear terms in one of the von Karman equations, a linear fourth order partial differential equation for the stress function is obtained. The particular solution to the stress function equation leads to multi·valued in-plane displacements, which are eliminated by proper selection of the homogeneous solution, The boundary conditions for the stress function equation are written in terms of the in-plane displacements, and wherever trigonometric functions of the small angle of rotation of the rigid inclusion appear, they are expressed in a power series of the angle and terms of higher order than the second power are neglected.

By using the resulting stress function and the Reissner solution for lateral displacement to evaluate the nonlinear terms in the second von Karman equation, a linear, fourth order partial differential equation for the second approximation to the large deflection lateral displacement is obtained. Again the boundary conditions are expressed in a power series of the rotation angle and terms of higher order than the third power are neglected. The solution for the lateral displacement is a function of the first and third powers of the angle of rotation, where the part containing the first power is the Reissuer solution and the part containing the third power is a correction term reflecting a reduction in lateral displacement caused by the in-plane stresses. Thus by neglecting the third power of the small angle of rotation, the large-deflection solution reduces to the linear, small deflection solution.

Any further iterations are not performed because the algebra involved becomes excessive; however, the iteration procedure can be repeated to obtain higher approximations. By taking appropriate derivatives of the stress function and the lateral displacement, expressions for the bending and membrane stresses as functions of the position in the plate and the angle of rotation are obtained. Numerical results are presented in graphical form for typical plates.

Experimental data was obtained with an 18 inch outer diameter, 7.2 inch inner diameter, 0.0634 inch thick plate made of 7075-T6 aluminum. The results of the iteration solution are found to agree very well with the experimental data for lateral displacements up to one and one-half times the thickness of the plate, but the iteration solution begins to overestimate the nonlinear effect for larger displacements.

As a limiting case to the title problem, an iteration solution for large deflections of a clamped circular plate loaded by a central concentrated moment is given.