45 



59 



shapes are compared in Figure 

 24. For a rectangle w, and Wj 

 on a side, deflected into the 

 shape characteristic of mem- 

 brane vibration in the lowest 

 mode, so that the deflection 

 at any point is z sin— sin:^^ 



Sphere 

 Poroboloid 



Figure 2k - Curves Illustrating Four Types 

 of Diametrical Profiles for a Diaphragm 



for small z. Thus for circle, cone and rectangle, respectively, for small z, 

 E "^ nahz'^, E=^7Tahz2, E = ^ n^ ahl^ + —) z^ [70a, b,c] 



For a square or w^ = W2, E is w/k times as great as for a circle at the same 

 central deflection. 



A correction for the elastic range is easily made, if required, 

 provided it is assumed that the elastic constants are unaltered by plastic 

 flow and provided resistance to bending may be neglected. During deformation 

 up to the elastic limit the area will increase by a definite amount AA^. 

 Since the stresses are at each instant proportional to the increase in area 

 up to that instant, the average stress will be a/2 and the energy absorbed 

 up to the elastic limit will be 



E^^fhAA, 



or half what it would be if the stress were constant. Thus, if E denotes the 

 total energy absorbed by the diaphragm, initially flat, up to a maximum in- 

 crease of area AA^, 



E = ^ahAA, + ah (AA„ - AA,) 



If AA is the residual increase in area after removal of the load, AA = 

 AA„ - AA.. Hence 



E = ah [aA„ - \aA,'^ = ah {aA + \ AA,\ 



[71] 



In general, the increase in area is proportional to the square of 

 the central deflection, for small deflections. Hence, if the central deflec- 

 tion Is z^ to the elastic limit, z^ to the maAimuri mder full load and z for 

 the permanent set, from Equation [71] 



z2 = zf 



iz^ 



zi = z"" ^z] 



[72a, b] 



