Bending of Thin Shells. 499 



G 2 can be expressed in terms of the changes of curvature, 

 and also that in the special cases alluded to a sufficient 

 number of the ten stresses are zero to enable the remainder 

 to be determined by means of the general equations of equi- 

 librium. 



5. We shall now determine these couples, using Thomson 

 and Tait's notation for stresses and elastic constants, and 



Love's notation for strains. 



Let A, B be two lines of curvature on 

 the middle surface of the undeformed shell ; 

 0,, 2 the centres of principal curvature ; 

 let oa, oh be the curves in which the 

 planes A O l5 B 2 meet any layer of 

 the shell. Let p ly p 2 be the principal radii 

 of curvature at 0, let Oo = 7j, and let 2A be 

 the thickness of the shell. Also let accented 

 letters denote the strained positions of the 

 various points. 



If P denote the traction along oa, and R 

 the normal traction along Oo, 



where 



Now 



and 



F = (m + n)<r 1 + (m — n) (<r 2 + cr 3 ) 

 = 2n{(l + E)o- 1 + Eo- 2 }+ER, . 



E = (m — n) J (m + n) . 



Fiar. 2. 



0) 



o-i = 



oa 



= 1 + 



V 



OA pi 



Since we neglect 



7 A'"" i+ / o 1 " 

 the extension 



of the middle surface, 



0'A' = OA, whence 



1 1+v/pi 



Similarly, 



" (k ~ ft) 



Pi' 



R 



m + n 



E^ + o-s)} 



\P2 P2 J P2 Ivi + n v v 



The value of G 2 is 



G, 



=f 



Trjdi]. 



(2) 



(3) 



(4) 



