the panel-buckling and general-instability problems. Both investigators 

 used the following buckling deformations: 



— mirx 



u(x,e) = A cos nS sin—- 



^b 



v(x,e) = B sinnQ sin ^^^^^ (172) 



Lb 



— mtrx 

 w(x,e) = C cos nS sin 



^b 

 i. e. , it is assumed that the cylinder will buckle into m half waves in the 

 longitudinal direction and n full waves in the circumferential direction. 

 Further, the ends of the cylinder are assumed to be simply supported. 

 The energy integrals for the shell and the ring frames and the in- 

 tegral for the work done (used by both investigators) were somewhat more 

 complicated than Equations (138) through (142) because of additional terms 

 that enter as a consequence of the out-of- roundness w . These integrals 

 are not given here, but the reader is referred to Equations (4} through ( 8) 

 in Reference 60. Once the total potential energy of the elastic system 

 (see Equation (137)) is determined, the minimum energy criterion leads 

 to a system of three linear, nonhomogeneous, algebraic equations which 

 must be solved simultaneously; these are of the following form: 



a^^A-t- aj2B + ^i^C = -ai4Co 



^12^"^ ^22^ "^ ^23*^ ^ "^24*^0 ^^'^^^ 



^13^''^23^''^33^ = -^34^o 



110 



