ne 
K = 8 ab eee eve eee eee (60) 
8h (a2 + b2) 
ee 2 
of = _il_ 80 cilbiprsyesabat eee eee eee eee (61) 
The final solution given by equations 58, 59, 60,and 61 will be valid so long 
as the plate is stretching, which is true up tot = W/op At this instant 
wWo(t) = wo =K and the plate is at rest everywhere with no velocity and a 
deflected shape w at any point (x,y) is given by 
w =K cos LM cos MLZ SOOO. .o0G. cog. (ee), 
2a 2b 
This represents permanent deformation and the plate will in fact remain in 
equilibrium thereafter since the applied pressure po cos ff x/2a cos fi y/2b 
is insufficient to cause further stretching. Equation 62 with K defined by 
equation 60 is therefore the final plastic dishing produced by the suddenly 
applied pressure p, os 1! x/2a cos 7! y/2b. 
150. The relative simplicity of the preceding example lies in the fact that 
the plate deflects in a constant shape throughout the motion. Similar 
results can be obtained for a pressure distribution of the more general normal 
mode type, namely, a pressure distribution of the type po cos m Ti x/2a cos 
n ti y/2b where m and n are any odd integers. The resulting deflection is 
then of similar shape. Up to the time of maximun deformation these solutions 
for a plastic plate are separately identical with those for an elastic 
membrane, For the elastic membrane, the solution for any distribution of 
suddenly applied pressure can be simply obtained by superposition of the 
previous solutions for the separate normal modes, However, for plastic 
deformation the principle of superposition cannot, in general, be used, 
Thus, if more than one mode is involved in the elastic case the membrane 
will in general stop stretching and start contracting at one point in the 
membrane whilst it is still stretching at another point. This introduces 
no difficulties if the strain is elastic since the same equations hold both 
for an increase and a decrease in stretching. For the plastic plate this is 
not true, the soap bubble approximation being only valid when the plate is 
stretching in both directions; if the plate ceases to stretch it may remain 
inextensible or tend to compress in one or more directions and the 
fundamental assumtions and equations must be changed. For any general 
type of impulsive loading on a plastic plate, it would thus be necessary, when 
seeking an exact solution with any given plasticity assumptions, to watch, 
in effect, every point in a plate and change the basic equations locally 
whenever the plate stopped stretching. Such an allowance for the difference 
between loading and unloading is complicated enough in the simpler case of a 
plastic wire subjected to dynamic loads; for the plate problem it has yet to 
be oarried out successfully. 
151. The difficulty introduced by the irreversibility of the plastic 
deformation can be avoided by further approximation. For the previous 
special gases of loading by pressure distributions of the normal mode type 
p,cos mi x/2a cos ni y/2b the simplicity of the analysis depends 
essentially on the fact that the plate deflects in a constant shape throughout 
the deformation so that all points of the plate cease to stretch and become 
at rest sinmltaneously. The irreversibility of the plastic deformation 
Sioply implies that the plate remains in this defornsed state without further 
motion. This suggests as an approximation for the case of loading by an 
underwater explosion that the plate be assumed to deflect in sone constant 
Shape. This assuiption has been termed proportional motion since the 
deflections of any two points remain in constant ratio throughout the motion. 
Thus for a rectangular panel uf plating of sides 2a, 2b, and thickness h, 
