The ring-energy integrals, Equations (153) and (154), are evaluated 

 at X = rL£ for the first summation of Equation (152), and at x = RL-p, for 

 the second and third summations of Equation (152). 



The shell-energy integrals, Equations (138) and (139), remain the 

 same for the present problem and so do the membrane forces in the shell 

 prior to buckling, i.e., 



pR 

 N^x=-i- (157) 



pR Lf h 

 ^oy = -^^:^ (158) 



The justification for using Equations (155), (156), and (158) instead of the 

 analogous expressions of Kendrick is discussed in Reference 58. 



With all this, the total potential of the system comprised of the shell 

 and ring strain energies and the work done by the external loads is given 

 by 



V^ = Ug+ Ub+ Vrings+ W (159) 



v/here Ug, U^, Vj-ings> ^^^ W are given by Equations (138), (139), (152), 



and (142), respectively. 



57 

 The*assumed buckling displacements used by Kendrick for the 



present problem are given by the following expressions: 

 u(x,8) = A,u,(x)cosne 



v(x, e) = 



sinT^ — + B3 sinTj^ — 

 Lf I -^1 Lfl 



sinne (160) 



100 



