4E h/E 

 s_ / t 



3R 



1 au 9v 



T — + — + ^ 



2 ax as R 



-^ + — + - + D 



E / 4 



t/1 a w 



E \4 4 



ax 



4 4 



aw aw, 

 ax as as 



4 



a w 



i a w 



+ ?— -^ 2 2 



9x ax as 



2 2 



, ^, a w ^, a w 



+ N — - + N — - + p = 

 x 2 s 2 



ax as 



(100) 



where x and s are, respectively, the axial and circurxiferential coordi- 

 nates and N^ and Ng are forces per unit length in the axial and circum- 

 ferential directions, respectively. 



By following a procedure similar to that used by Donnell, Reynolds 

 was able to combine Equations (98) through (100) into a single eighth- 

 order equation in the radial displacement w only; this result is given as 



fE „ / E 



4 4 



,^4/ 3 a w aw 

 ax ax as 



E h 4 



r' ax^ ' 



4/ a"w _ 3^/ _s\ a w 



2 41 ~E ) 6 

 ax \ t / 3x 



3^/^ s\ / 3 aw a'w 



4\ ~E jU 8 ,62 



t/ \ ax ax as 



as / V t/ ax as 



= 



(101) 



where v indicates the operator f -2 — ^ 



2 \2 



2 2 



ax as 



Reynolds assuined the following buckling shape as a solution to 

 Equation (101): 



w(x,s) = A sinks sinXx (102) 



67 



