The further simplification which was effected in Z by the assumption of 

 only small values of p cannot be used here since the ratio of the radius to the 

 frame spacing is sometimes too large. On the other hand, in most calculations of 

 submarine hulls we have to do with a larger number of lobes, n. It is possible, 

 therefore, if n equals at least 8 or 10, to neglect the last two terms in the 

 brackets of Eq. (6) in comparison with the first term (n 2 + fit- 2 ) 2 . In the same way, 

 we may neglect 1 in the denominators in comparison with n 2 , by which the error made 

 by the previous assumption is partly compensated. We have, then, with sufficient- 

 ly close approximation for all cases of practical importance, the final formula: 



y -^77Z [(;y^' «--»*> ♦<»■*««■ 



X 



• | \c*r i- or/ 

 2 



This is even simpler than the equations (C) and (D) in Z. If we substitute here 



as before (T = 0.3, and further for x and y the values from Eq. (6'), we obtain 



for the required buckling pressure p, if 2a designates the diameter, 2h the wall 



thickness and 1 the effective frame spacing: 



(7) 



_ E h 

 P „2 ^ #? a 



S 2 + %.a )5 + 0.73 (n 2 + * 2 ) 2 h 2 /a 2 



(8) 



where 0t - Ta/1 



For n use that whole number which makes the expression for p a minimum for 

 a given a, h, and 1. E designates the elastic modulus of the material as long 

 as the stresses in the shell are below the proportional limit. Concerning the 

 applicability of the equations above the proportional limit, similar conditions 

 apply as in the case of ordinary buckling- of rods (see below). 



Eq. (7) and (8) do not give accurate values for a tube of infinite length 

 since here, as is well known, the number of lobes, which we have assumed to be 

 large, decreases to two. 



2. DISCUSSION of the FINAL FORMULA. TABLES. 



Eq. (6), as well as the simplified Eq. (7), represents a straight line in 

 an x - y coordinate system. Each wave number, n, represents a straight line for 

 a given ac/1, as shown in Z. Hence, y as a function of x will be represented by 

 the ordinates of a straight line polygon. The vertexes of each one of these pol- 

 ygons increase in number towards the origin of the coordinate system. 



The significance of °c requires more detailed explanation. According to Z, 

 Eq. (12), 



OC= TTa/1 (9) 



The "effective" length of the tube is to be substituted here for 1; that is, the 



