_ 3 - 109 



point from its initial position in a direction perpendicular to tht sheet, at any rate so far 

 as any one state of the di-ipnra^m is concerned. When the strains at a jiven point in the 

 diaphrajm are compared at variojs stages of displacement they are proportional to h^ . 



The mean displacement is 



- _ Volume, V , Between initial and final positions of diaphragm 

 Area, A, of diaphragm 



For a circular diaphragm and small displacement 



Ellijitical and other non-circular diaphragm s. 



(14) 



Though the distortion of a non-circular diaphragm cannot Oe treated so simply as that 

 of a circular one, an approximation to the displacement might be made by assuming that the 

 diaphragm assumes the same form as a flat membrane or soap film when displaced by a uniform 

 pressure perpendicular to its plane. There is, howuver, one non-circular shape for which the 

 complete calculation of stress, strain and displ'-icement can be made, namely for elliptical 

 diaphragms. This calculation is here carried through in order to estimate the error that may 

 be expected if the assumption is made that the diaphragm Is displaced into the same form as a 

 memOrane with uniform tension in all directions. 



Taking the equation for the edge of the elliptic diaphragm or plate as 

 „2 ..2 



X 

 72 



b- ' 



(15) 



where 2a is the m-ijor axis, it will be assui'id that the equation for the displaced sheet is 



•^ = 1 - 

 "o 



x2 



^ 



(16) 



and It will be shown that this assumption is consistent with the satisfaction of all the required 

 plastic stress, strain and equilibrium conditions. It will be assumed provisionally that the 

 stress is uniform at all points but not isotropic, tnus ct and a will be taken as the components 

 of stress parallel to the surface of the plate. The condition of equilibrium In direction normal 

 to the surface is 



p = t 



CT 



(17) 



Where t is the thickness of the plate and p^^, p^ are the principal radii of curvature. For a 

 sheet of the form (16) i/'p 



2h^/'a% l/p = 2h^/b2 so that 





(IS) 



since (l8) does not contain x or y it can Oe satisfied if, as has been assumed, a and a are 

 independent of x and y. It will be noticed that (l«) shows that the displaced plastic sheat is 

 In fact of the same shape as a displaced soap film. The fact that a =0" In the soap film, but 

 not In the present case, rrakes no difference to the form of (16). 



It now remains to find out whether a distribution of displacements and strains in the 

 surface of the sheet can be found which is consistent with the noriral displacement (16) and at the 

 same time allows the plastic stress-strain relations to be satisfied. 



Plastic stress-strain relationships. 



Assuminq that o-^^^ is constant over the ellipse and equal to a, it seems clear that, 

 whatever the stress-strain relationships may be, the ratio of the strain components In the surface 



of the 



