288 - 6 - 



that, during the time of movement of the piston and sprino representing th plate, plastic *av3s 

 from the edge have time to reach the centre of the plate. If this is not so, it is clrar that 

 the model is not fully representative, so the point must b? checked for c-ach model that we use. 

 The test will be more stringent for the "box.mocel" case where the only clanping is at the .idgcs 

 than for the "ship" case where the plastic wives havt only to travel from th' stlff^ners to the 

 cc-ntrs of each panel . 



(3) Comparison of various models for pi ites cf large radius . 



We shall carry out all the actual calculations with the "ship" mode), which will be a 



2 ^ 



panel of equivalent radius " divided up by stiffeners into N panels each of equivalent radius tt. 



We shall suppose that each of these sub-panels can be represented as a piston on a spring. It 



would be a better approximation to regard each sub-panel as deforcing into a parabolic snipe, as 



we could then regard the stiffeners as being at rest, but this leads to difficulties when we 



are considering the models in which cavitation occurs. Our experience in the past has been that 



changing over from "piston" to "paraboloid" approximations does not alter the order of' magnitude 



of the results (See ll). In this report we are only trying to find approximate criteria for the 



effect of cavitation on damage, and the "piston" approximation should be satisfactory for this. 



We need an expression for the strength of the spring per unit area of plate. A circular 



o 



plate of radius - when deformed into a cup of a 'phere or paraboloid so that the mean deflection 



N 2 2 



is y increases in area by i tt y . This corresponds to a total plastic energy o u cr h tt y , where 



2 2 

 c Is the yield stress and h the thickness of tht plate, corresponding to an energy of n a f^^ y 



per unit area of plate. Thus, to represent thi plastic forces we need a spring of str-ngth 

 2 

 ^0 per unit mean deflection per unit arec'.. We shall use this in all subsequent calculations. 



To go over to the "box-model" we simply nut N = i and allow for the effect of the baffle as well 

 as we can. 



(3-l) Incompressible Theory . 



The investigation given in II has shown us that incompressible theory is probably a good 

 approximation for small platc-s, so it is of interest to set- what results it gives for large R. 

 Tne equation of motion, y in all cases representing mean deflection, is:- 



(^. ' i'°') dt 



2 8 N^ cr h ^ 



Ll . __^.^ y = p n-9 (5) 



where p is the maximum pressure in the explosion pulse, its time-constant, p the density of 

 the plate and p the density of the water. Since the plates are all moving as pistons, the 



"will be the same as that for one large piston. For large R we may 







neglect ph compared with this term. The solution of equation (5) 



8p R {i *v e^ 



t 



p-d + sin \ t _ cos \ t 



3 TTCr^^ N^ 



— T^^r- 



where K 



2 _ 



(6) 



R 

 ke suppose R and N to become large in such a way that -n remains ^ixed, corresponding to a fixed 



size of sub-panel. X. then becomes small, and the middle term in the bracket large compared v/itn 



the other two. The maximum deflection is given by;- 



>"" 8 N ip^ cr^ hi* 



thus tending to zero for large R. If, however, we take the "box-model" assumptions \ again becomes 

 small fof large R when we put N = 1 and the maximum deflection is now given by: 



