throughout. This latter is an approximation since Ngy is really a func- 

 tion of X as can be seen from the theory of Von Sanden and GUnther, or 



g 

 from that of Salerno and Pulos, but it can be shown that the error intro- 

 duced by this approximation is negligible. 



The extensional and bending strain energies, respectively, of a ring 

 frame situated at a distance x = rLf from a bulkhead (one end of the 

 cylinder) are given as follows: 



2 r'' (140) 



^^•OQ-'^^I -(^'0-^^ 



^Jll^ + -^ f'0+-'9-^"'0^)^^^^^^ 



Ztt 



o f 



where Nq£ is the hoop thrust in a ring frame just prior to buckling and will 

 be assumed to have the value -pRLrAr/IAf+Lfh) instead of, say, that given 

 by Equation (21). The quantity I^ is the moment of inertia of a ring- 

 frame section about the x-axis, and e is the eccentricity ( + inward) of the 

 centroid of a ring-frame section. 



The potential due to the radial pressure p, involving the additional 



displacements u, v, w, due to buckling is given by 

 2tt 



o o 



The work done by the axial pressure p becomes equal to zero as a conse- 

 quence of the trigonometric functions assumed for the buckling 



87 



