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.
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