Response of a plastic circular plate to a distributed time-varying loading

From the results and equations shown herein, several important conclusions are evident. The equations derived here considering bending deformations only are seen to be more general in form than existing solutions, and reduction to the existing cases is direct. For example if the loading is considered uniform in r and impulsive or step-wise uniform in time, the equations derived directly for such cases by Hopkins and Prager and Wang (refs. 2 and 5) appear exactly. Also, if the radial load distribution is considered uniform, and a general function of time is allowed (but assuming only inward hinge circle movement), the nonlinear equations of Perzyna (ref. 57) are found exactly. The conclusion of Perzyna that time variation is unimportant appears to be caused by an unfortunate choice of example time functions. He solves the specific non-linear equations for his example, and does not present any means for evaluation of his numerical method of solution.

If the loading on the plate is considered to be a distributed Gaussian loading in r and impulsively applied, the equations derived directly for this case by Thomson (ref. 56) appear exactly herein. These two papers (by Perzyna and Thomson) are the only two papers available at present that allow variations of the loading, one in r and the other in t, and both sets of equations are included in the general expressions herein. In fact, the solutions currently available for bending theory are found to exist as special cases of these general equations.