280 Cilley — Fundamental Propositions in the 



If we compare these with the strains of the ordinary solution 

 we will find them much more constant except for the axial 

 strain which is zero in the ordinary solution. This will be 

 most apparent if we consider the primary strains which, added 

 to the strains of the ordinary solution, give the strains of our 

 case. 



We find that the primary stresses are* a radial compression 

 diminishing from the interior layers of the ring in proceeding 

 towards either surface, a circular stress, compression at the 

 inner surface, tension at the outer surface and an axial stress 



(T li+P Pi r t / „ rj. r \ 



— — a — - — = — ( 1 — a — 2<x 2 + a — ) tangential strain 



E E E r —ri \ r + r 1 r J ° 



— actually extension) 

 P+Q pi \ r^o 



f=~° 



q = — — a 

 y E 



+ Q pi \ r x r a /r +ri a . . 



—=— =a — — i — 2) (axial strain — varying from 



extension to contraction) 

 At the inner boundary r = r x these take the values 



— Vi / V\ ?*i 2 \ 



«i = — ^r( 1 + (T •" 2 ff2 — o ., ) (radial strain — actually contraction) 



— Pi / r i f" 1 \ 



f 1 =z—[ ye — 'la 1 — ) (tangential strain — actuary extension) 



E \r — 7"i r ~ — 7i / 



g\ = o -^- — - — (axial strain— actually extension) 

 E TQ-\-r x 



At the outer boundary r = r they take the values 



— Pi f *"i ^*i 2 \ 



e = — tt ( a + 2(y2 — t, 7T I (radial strain — actually contraction) 



E \ ro—T! r 2 — n 2 / 



— P\ ( r i r \ i \ 



fa = — ( 2a' 2 — - J (tangential strain — actually extension) 



— Pi f\ 



go — — s — (axial strain — actually contraction) 



E ro+r-L 



We see that the radial strain at the inner surface exceeds that at the outer sur- 



Vi 

 face by — — -, the tangential strain at the inner surface exceeds that at the 



Vi 

 outer surface by o^—, and the axial strain at the inner surface exceeds that at 

 E 



V i 

 the outer surface by cr^— , these latter being of opposite signs and nearly equal 

 E 



amount. This all corresponds to the fact that the difference in stresses at the 



inner and outer surfaces is simply the compression p Y at the inner surface. The 



law of variation of these strains is in each case simply hyperbolic (varying as the 



reciprocal of the distance from the axis plus or minus a constant). 



* The primary stresses are 



r r x (r— r ) (r— rj 

 r 2 (r 2 -»i 2 ) 



(P) = P — P = pi 1 ., ." — - — - — ■ (radial stress — always compression) 



(Q)=Q— Q = Pi — r— rf- (tangential stress — is tension or compression 



r\r 2 — ?v) > \ 



according as r 2 <--• r ri J 



(R) = P — P = (axial stress — nothing). 



