59 
4154. The preceding discussion indicates an approximate method of treating 
the plastic dishing of a plate under a given dynamic pressure. for the 
dishing of plates by underwater explosions the pressure on the plate is not 
known, ab initio, sinoe as previously discussed it depends on the motion 
of the plate and, in particular, on whether cavitation ocours in the water. 
The problem is then more complicated, but the assumptions of proportional 
motion and uniform membrane stress have, in general, been the main simplifying 
assumptions (so far as the plate is concerned) in the more complicated 
analyses“allowing for the interaction of target and water phenomena, These 
analyses lead to ths conclusion that in many cases, the pressure puls@ from 
explosions tend to communicate a certain definite quantity of energy which is 
approximately equal to the energy directly incident on a deforming target 
plate. If this is so, the final dishing of the plate oan be estimted 
without intermediate calculation of the actual motion, Thus using an 
assumed deflected shape of the type of equation 64 the total energy absorbed 
in plastic deformation is given by equation 65 and hence, by equating energy 
absorbed to energy communicated, 
1 -& (a+) Me mapas osaseyl (9) 
where w, is the final central deflection, The directly incident energy 
depends on the charge weight and the geometry of the target and the relative 
position of the charge; if bubble damage is appreciable the contributions 
to NN, from the spubse quent bubble pulses must be added to that from the 
pressure pulse. 
155. Equations of the type 66 have been the most common simple basis of 
comparison of theory and experiment for box model and drum model tests, 
One minor point which may be noted is that for different assumed shapes 
£(x,y) of proportional motion, the energy absorbed varies less for a given 
mean deflection than for a given central deflection. Thus if W is the 
mean deflection, the energy absorbed can be expressed in the form 
Qip =4*'¥ ves’ eee, Seen eetn egy 
and if this is equated to the incident energy SL; an estimate of W instead of 
Wo is obtained, Such estimates of W vary less for different assumed shapes 
than do corresponding estimates of W. For this reason, results of box or 
drum model tests are often expressed in terms of the mean deflection rather 
than the maximum deflection. 
156. The use of an assumd shape for calculating the final absorbed energy 
p in the preceding manner does not, of course, depend on the assumption of 
proportional motion, but only on the more general assumption that ths plate 
stretches throughout its deformation so that the uniform membrane stress may 
be assumed as a reasonable approximation, fhe assumption of proportional 
motion is, however, of importance since it ia involved in analyses forming 
the thspretical basis for the conclusion that approximately all the incident 
energy\ 24; is communicated to the target. 
157. The preceding type of analysis is especially relevant to experimental 
investigations using box or drum models embodying a single plate fixed ina 
relatively rigid frame. For an actual ship the supporting framework of the 
panels cannot necessarily be assumed rigid and the basic system is rather a 
panel of plating supported by yielding beams, This more difficult problem 
is likely to defy exact analysis for a long time but using an approximate 
method the deflection of any section of the beam and the deflections of the 
ends of the beams can be estimated, This method is of general application 
to problems of dynamic loading of structures but is especially relevant to 
the present type of problem since it enables the difficulties introduced by 
the irreversibility of plastic deformation to be overcome. 
