cylindrical shell supported by ring frames of finite stiffness; this is dis- 

 cussed more fully in Reference 8. 



It can be shown that the general solution of Equation (1), which is a 

 linear, ordinary differential equation with constant coefficients, can be 

 written in the form 



w(x) = A sinhX,x + Bcosh X, x+C sinhX-X + Fcosh X,x-^(1- v/Z) (2) 



1 1 3 J -tyn 



The characteristic roots \-^ and X^ of the fourth degree auxiliary equation 

 which results when w(x)'«'e'^''' is substituted into Equation (1) are given by 



^-^-^^-(7)^(^) 



-1 



1/2 



1 1/2 



(3) 



w^here 



and 



e = \| 3(1-1/2) 



\lRh 



* _ 2E(h/R)' 



^3(1-1/2) 

 is the critical load for the axisymmetric elastic buckling of an unstiffened 



cylindrical shell under the action of uniform axial pressure (see Refer- 

 ence 9). 



An examination of the roots Xj and X3 reveals that four possibilities 

 for the ratio p/p exist and that they all influence the nature of Equation 

 (2) . These are 



(4) 



(5) 



20 



