that the cylinder buckles into a half- sine wave in the longitudinal 

 direction. It can be shown numerically that lower values of p^^ result 

 for m = 1 than for m > 1. Physical intuition also leads us to accept this 

 because the buckling shape corresponding to m = 1 is associated with 

 lower energy states than those for m > 1 , at least for the case of hydro- 

 static pressure loading; however, this may not be so for the case of ajxial 

 loading. 



An examination of Equation (57) reveals that the critical pressure is 

 dependent on the value of n. This means that for a given geometry of 

 shell and for a given material, calculations must be conducted for dif- 

 ferent values of n in order to find that value of n which minimizes the 

 pressure. It is this minimum pressure one seeks. To facilitate this 

 calculation process. Von Mises developed a set of curves; these can be 

 found in either Reference 30 or Reference 32. 



Another approach to this minimization is to do it analytically and 

 thus find an expression for p^^. which is independent of the parameter n. 

 Windenburg and Trilling did exactly this in Reference 32, and the final 

 convenient formula they arrived at is given by 



2.42E 

 3/4 

 (1-2/2) 



(h/2R)^^2 



J±. -0.451 Jl\^fz 

 2R \2Ry 



(59) 



Calculations carried out for a range of L/2R from 1/8 to 2 and a range of 

 h/2R from 0. 002 to 0. 007, for assumed values of E = 30 x 10^ psi and 



49 



