where k = — ; \~ — = — ; m and n are integers which denote the number of 

 half-waves in the longitudinal direction and the number of full waves in 

 the circumferential direction, respectively. Equation (102) satisfies the 



conditions of simple support at the ends of the cylinder, i.e., that w and 



2 

 9 w 

 -vanish at the ends x = and x = L. These conditions are not un- 



3x 



reasonable for stiffened cylinders since it is likely that the effective 



rotational restraint will be limited by the formation of plastic regions 

 arising from high bending stresses near the ring frames. 



By substituting Equation (102) into the differential equation (101), 

 Reynolds obtained a characteristic-value equation, which using the follow- 

 ing notation 



x^k^ i^iJ^Y 



\mvl{) 







f =^ = -^ (103) 



p 2N^ dg 



E 

 t 



and rearranging terms to solve for the plastic buckling pressure Pp, leads 

 to 



- X X nl E hf ^^ 

 P = p— = ,, , , /- S_i. ,104, 



2f D/.^ :F"y"^' 



R+[l-*(l-g] [l + lc* ] 



68 



