C4 z \2(n2+x2)2^ ^2^2 



X _ rrnrR 

 L 



L - Lr-b 



(56) 



Assuming that the shell is thin and keeping only the principal terms in 

 Equation (55), Von Mises obtained the following simplified formula for the 

 critical value of the pressure: 



Eh 

 'cr = R 



Ln2+l(TrR/L)2_ 



C^fhY 



(h/R)' 



:^^(^i 



H57) 



[n^(TiR/L)2]2 12(1-1/2) |_ 



It is important to point out that the above equation results as a conse- 

 quence of the assumption that n is large, say on the order of 10, so that 

 then (n^-1) =? n^ . This implies that Equation (57) is not accurate for very 

 long shells which buckle into the elliptic shape, i.e., n = 2, or even for 

 relatively long shells which may have a critical buckling mode correspond 

 ing to n E 3, 4, or even 5. For such cases, which are not usually encoun- 

 tered in ring- stiffened cylindrical pressure hulls, recourse must be made 

 to the more exact formula developed by Von Mises; see Equation (6) in 

 Reference 32. For the limiting case of a cylinder of infinite length, i.e., 

 L->oo , which buckles in the oval shape as does a "free" ring under radial 

 loading, the following simple formula of Bresse and Bryan (see Reference 



32) is applicable: 



E /h\3 



^""' 4(1-1/2) IrJ 



(58) 



Furthermore, Equation (57) is based on the fact that m :; 1 which implies 



48 



