110 Dr. J. Prescott on the Equations of Equilibrium 



Thus P 1 = ^(a 2 -rO, (33) 



r 2 =^ 2 -30 (34) 



Since V 4 w=0, 



equation (28) gives 



_ 2AE d It d<f>\ 

 ^ r dr \c dr ) 



c 4c v ; 

 = ^^(2r 2 -a 2 ) (35) 



4 



G 



It should be observed that the total thrust on the plate, 

 namely, 



I lirrpdr, 



Jo 



is zero, and also that the pressure p changes sign at the 

 circle where 2r 2 = a 2 . Moreover, the circumferential tension 



P 2 is positive where r is less than— -7^, and negative, that is, 



a thrust, where r is greater than —777. 

 The mean shear stress ¥ 1 is 



„ 1 EA 2 d ,_ 2 . 



= (36) 



The bending moments per unit length along and perpen- 

 dicular to the radius are 





_ 2EV_ (37) 



- 3(l'-<r)«' 



M 2 =Mj • • • • (38) 



We thus find that [the middle surface o£ a circular plate 

 will take the £ormj of a portion of a spherical surface of 

 radius e provided that a constant bending moment given 



