where p^.^. is the buckling pressure of the perfect cylinder and is given by 



Eh/ 

 ^cr = "J 



2ttR 



A 



-, 2 L 2 2,2 



,(-hM^(3„S2„^i+A'»)l (126) 



and A = —^ — • The smallest value of the buckling pressure p is found 

 L, cr 



by minimizing Equation (126) with respect to n. A similar relation for 

 the elastic buckling of a cylinder with clamped ends was derived by 

 Nash using an energy method. Bodner and Berks derive a relation 

 similar to Equation (125) for simply supported imperfect cylinders. 



Combining Equations (123) and (124), we obtain the following expres- 

 sion for w: 



r(x,e) = -( — 2 — \ sinne 

 2\Pcr-P/ 



1- cos 



2lTX 



(127) 



The bending moments in the shell can then be calculated from the 

 w^ell-known relations 



M^=-D 



Mq = -D 



W'xx+ — ^'99 



] 



ri 



,+ v 



^.09^" W'xx 



] 



(128) 



The maximum bending stresses are then given by 



"^bx =± ^(^x)max. 



^b9=±^(Me)^ 



(U9) 



To obtain the total normal stresses, we have to add the membrane stresses 



78 



