element of a cylindrical shell subjected to the simultaneous action of axial 

 compression (N^) and uniform lateral pressure (q) are given by 



2 9^^ U|. 3^^ 

 2 2 ax80 



ax 



3w / a V aw\ 1-1/ a u _ 



ax "^ **'i\ax90 ~ ax / ^ 2 ^„2 ~ 



2 2 2 



1+1/ a u , i-i/-^ 2avav aw- 

 ax ay 



2 3 ^3 



a V , a w , -o2 a w 



__+__ +R — ^— + 



ao ao ax a9 



+R^l-^)^^ 

 9x J 



ax 



^ au dv — 



1/R — + —r - v/ - o( 

 3x 30 



^ + (2-i/)r' 



a0 



3 ,44 



9 V ^4 9 w 9 w 



— r— + R — 7+ — 7 + 



2 4 „4 



9x 90 9x 90 



+2R 



2 9 w 



2 „2 



ax 90 



2 o 2 



/ 9 Wx , ^2 9 w 

 >^(w+— )+ct>^R — 



90 9x 



;52) 



where 



N 



h 



■ o(= - — - 



' 12R' 



(53) 



for the case of uniform external hydrostatic pressure, 6 -y = — cb , and 



"^ 2 ^ 



N. 



pR 



X - 2 • 



Equations (52) are linear and homogeneous so that a solution for the 



displacements u, v, and w in terms of sine and cosine functions is 



possible. Von Mises assumed the following buckling displacements which 



satisfy the conditions of simple support at the ends of the cylinder: 



46 



