130 



45 



P = 



4t2 



1(4] 



where a is the radius of the boundary of the membrane. 



A membrane deflected under pressure forms a spherical cap. The 

 volume enclosed by such a cap and the plane of its boundary is 



V = \na^z + \ttz' 



15] 



The energy tV absorbed by the membrane as Its central deflection in- 

 creases from zero to z is 



W 



= jpdV = j- 



^TZ 1 5 



+ z')dz 



:i6] 



* f 



T 



P re SSL 



For a thin plate deformed plastically under pressure the tension r 

 is given by r = a'h', where h' is the thickness and a' is the stress in the 

 plate, assumed to be of equal 



magnitude in all directions par- , 



allel to the tangent plane of 

 the surface at a given point. As 

 stated in the foregoing, t remains 

 approximately constant as the 

 pressure is increased. The value 

 of the constant for different 

 plates may be approximated in sev- 

 eral ways. In this report it Is supposed that the constant is determined by 

 the ultimate tensile strength a^ of the material, and the initial thickness h 

 of the plate according to the formula 



I Vv 



I — : 

 -° h 



Figure 36 - Diagram Showing Notation 



T = a a„h 



171 



where a is a constant for all diaphragms made of the material. 



Equation [17] may be based on a notion of affine* materials (l6) 



introduced later in this report. 



We may write, using Equations [l'^], [l6], and [17] 



V = 



4aa^h z 

 a' + z^ 



Dr. Osgood in References (l6) and (17) discusses stress-strain curves which are affinely related. 

 The concept is suggested here by the phrase "affine materials" by which is meant materials whose plas- 

 tic behavior, when referred to a given property of each material, reduces to the same form in all of 

 them. The property chosen in the present case is ultimate strength. 



