108 



- 2 - 



where r, Is the radius of the diaphrajm. The pressure necessary to produce the displacement is 

 p = UPth^/r^^ (5) 



where t is the thickness of the sheet. 



The work S » done on the material by the pressure during the displacement of the centre 



from h to h + 8 h^ is 



8 W = 



277 pr [^ 8 hj dr 



(6) 



Substituting for p and z from (u) and (5) and integrating with respect to r and h the total 

 work done is 



w = 77 Pt h. 



(7) 



Since 77 r, is the area of the diaphragm the average work done per c.c. of the material of the 

 plate Is 



P 1 



(a) 



Distribution of plastic strain in the dia'phrapm. 



Though (s) represents the average amount of work aone over the whole area of the diaphragm 

 tf\e distribution of strain, and therefore the work done per c.c, is far from uniform. If t, is 

 the radial component of displacement parallel to the initial plane of the diaphragm the two 

 components of strain are 



d$ 1 



radial, €, = + - 



tangential, e = J/r 



(9) 



Since O", = a, symmetry ensures that £, = £,. The equation for ^ is therefore 



2 

 ai I Idz 



dr 2 dr 



Ur 



(10) 



and substituting for 2 from (4) this becomes 



"C S ^K'r' 



dr r 



(11) 



The solution of (ll) which corresponds with ^ = at r = r, is 



i = h 



r^ 



2 ■ — 1 



(12) 



The principal strains £, and e. are each equal to 



K 2 



^ 



t-d 



(13) 



It appears therefore that the strain is zero at the edge and equal to h^j /r, at the centre. 

 The strain at any point in the surface of the sheet is proportional to the displacerrtnt of that 



point 



