qf = PfRoLe = (n^-l)-^ 



so that 



Pf = (n^-1) Y~~ (^^^') 



where Rq is the radius to the outstanding flange for an external ring- 

 frame, but however, is defined as the radius to the contact surface with 

 the shell plating for an internal ring frame; and R„„ is the radius to the 

 centroid of the combined cross section made up of one frame and an 

 "effective length" Lg of shell plating. The quantity Lg can be computed 

 using Equation (22). 



Just as in the case of Equation (57), it is necessary to minimize the 



critical buckling pressure p^j. of Equation (136) with respect to the number 



4-ft 

 of circumferential lobes n. To facilitate calculations. Ball ° has develop- 

 ed a graphical solution of Equation (136); further discussion of this will be 

 given later in connection with some of the more complete formulations of 

 the general-instability problem. 



The next major theoretical development for this problem was that of 



the group at the Polytechnic Institute of Brooklyn, notably the work of 



49 

 Salerno and Levine . Their method of solution was based on the principle 



of minimization of the potential energy as for the shell (lobar) buckling 

 problem. The same general system of energy expressions was used in 

 both cases; however, in the general-instability problem, the total energy 



84 



