112 



h-tP 



p = _2_ (a + 1//9) according to Mohr's theory (JO) 



a 



or p =■ -2t- (a + 1//3) (l -a + a^) according to von Mises' theory (31) 



If the stress is assumed uniform, as in a soap film, cr^ ~ O" ~ P so that 



X y 



h tP 



(32) 



For the case when D/a = 2/3, /9 = 1/9 and the curve of Fi^re 1 gives a = 0.86. In this 

 case the factors multiplyinj h^tP/a in (30), (3l) ana (32) are 3.H, 3.31 and 3.25 respectively. 

 The assumption that the plate behaves tike a membrane with uniform tension equal to P in all 

 directions therefore overestimates the work which is necessary to produce a given displacement by 

 4.5S if Mohr's criterion is accepted. On the other hand this assumption underestimates the work 

 by 1.81 if von Mises' strain energy criterion is used. 



As the length of the ellipse increases the error Increases if von Mises' criterion is 

 used. It is 6» when a/b - 2, lOJ when a/b = 3 and 15i when a becomes infinite. 



Rectangular plate dished by uniform pressure . 



Though the exact analysis which can be completed for the elliptic plate is very difficult 

 for other shapes, the approximate calculation assuming equal tensions in all directions can be 

 carried out for rectangular plates. For plates 6 feet x 1 feet, such as are used in the box 

 mooel trials, the error involved in making this approximate assumption is likely to be of the 

 same order as the error in making the same assumption for elliptic plates with axss in the ratio 

 3:2. This error has been seen to be about 21, 



If 2a and 2b are now the lengths of the sides of a rectangular plate, the volume V enclosed 

 between the distorted plate and the original plane is, on the above assumption. 



V = i { « ab3 } j I _ ip (|) [tanh ^ . ' tanh 1^ * ... ] 



Pt 



2b 



When b/a = 2/3 this becomes 



V = 0.232 pa''/Pt 

 The maximum displacement in the middle of the plate is 

 k2 



Pt 



16 b' 



fsech^ 



L 2" 



sech 



3 TT a 



♦ 5 -' sech 



5 TT a 



(33) 



(34) 



.. (35) 



and for b^ = 2/3, (35) gives 



h^ = 0.179 pa /Pt 



The mean value of displacement of tho plate is 



z = v/Hab = 0.0870 pa^/Pt When b/a = 2/3 



(36) 



z/h = 0.087./0.179 = 0.485 



h /z 



(37) 



work done on the plate (0/3 = 2/3) is 



5.76 MPt(i)^/a^ 



(38) 

 where . 



