114 _8- 



Uniform distribution of impulsive load. 



The foregoing discussion assumes that the velocity given to the plate is that which would 

 apply in a normal vibration of a membrane. For th9 rectangular plate this implies a distribution 

 of Initial impulsive velocity proportional to cos ^-^ Cos ^Jg^, i.e. it Is greatest in the middle 

 and falls gradually to zero at the edges. If a plastic plats is subjected to a uniformly 

 distributed impulse such as would be given by a very large pressure acting for a time which is 

 small compared withT the central p-^rt of the pl^te will go on moving with uniform velocity till 

 the transverse wave of material which nas stopped moving redches it. The velocity with which this 

 wave is propagated from ths edge Is c. For a circular elastic membrane the motion is complicated 

 because the outef part cjmes to rest before the inner part and begins to return to the equilibrium 

 position while the centre is still mjving away frjm it. With a circular plastic diaphragm, however, 

 this complication will not arise. If the ideal plastic stress-strain relationship is assumed 

 (1.8. no strain till the elastic limit is reacned and tnen constant stress with increasing strain) 

 a circular diaphragm dishes into a circular cone. The angle which the generators of this cone 

 make with the plane of the undistorted diaphragm is ff = v/c, where v is the initial velocity given 

 to the diaphragm by the Impulse. A similar simple solution might perhaps apply when the plate 

 is rectangular. If such a solution is possible the central flat area will reirain flat and 

 parallel to the original position of the sheet while its area diralshes with time owing to the fact 

 that tha sloping sides are propagated into it with velocity c. The final "dished" shape of the 

 rectangle Is like the f'oof of a house whose plan form Is rectangular. Contours are shown In 

 Figure 3. 



The contours of the dished plate for constant values of z/h are the same as those shown 







in Figure 2 for the displacement which results from that distribution of impulse which corresponds 

 Kith the slowest normal mode of vibration of a membrane (i.e. z^h = cos ^r^ cos ^ji^ ). It is 

 not suggested that a real plate would actually be displaced into such a shape, the properties of 

 steel are not those of the ideal plastic sheet contemplated, but it might be interesting to measure 

 some "dished" steel plates to see how near they are to one or other of these shapes. 



The ratio z/h for the shape shown In Figure 3 is 



I = 3 ^' * ^ (3) = "-J? or^ = 2.57 (47) 



PART 2. Compari son of box model expcrimi'.nts with 

 theoretical di scussion . 



In the box model experiments steel plates approximately 6 feet x » feet were bolted round 

 the edge of a neavy iron box so that they formed one side of thd box. The plate was thus air-backed 

 and supported round its edges. The box was towered to the required depth in water and explosives 

 fired at various distances. In g3neral they were situated on the normal to the plate through its 

 mid point and at distance X from it. The principal measjrements made in cases where the plate 

 did not burst were the volume V contained between the dished (late and its original position and 

 the maximum displacement h^. The mean displacement z is evidently (if the plates are 6 feet x 

 U feet) 



7 . JL _ Volume in cubic inches _ V . ^ ,,.., 



z = nTh = - TOSi inches (48) 



tab 4 X 6 X 144 ^*^' 



The mean displacement or deflection of the plates, z', was also estifieted by measurement and 

 tabulated. On comp,irison with the value obtained using (48) it was found that In all cases 

 z'/i = 1.07. This discrepancy can be accounted for if the area of the plate Is 7« less than I', 

 assumed. This error can be explained if tne plate is really 5 feet 10 inches x 3 feet 10 Inches 

 instead of 6 feet x 4 feet. If a 1 inch border is rendered imriobile by the arrangeirents for 

 gripping the edge, this discrepancy is removed. Assuming this to be the case we nay now write 



- V V 



' ' VC - U^ - 3275 '"^''^^ (*8a) 



Value 



