N is the total number of ring frames on the cylinder. 



These quantities, as used in Reference 51, are defined by the following 

 integrals in terms of the displacements u{x,9), v(x,8), and w(x,e): 



U = 



ERh 

 2(1-1/^) 



2 2 2 



u, +(v, „-w) /R +21/11, (v,„-w)/R + 

 X X 9 



^4^ (v.^+a,g/R)^ d0dx+ ° 



,22 



(u, Q+v, - 2v, qw) d9dx + 



2 2 



(v. +w. ) dQdx 



X X 



138) 



\ = 



2tt 

 24R(l-v )j J ^ 



2 2 2 2 



R w, +(w,-„ + w) /R + 2l/w, (w,„_+w) + 

 XX 09 XX 09 



+ 2(l-l/)(w.^g-f^v.^-^a.^^' 



d0dx 



139) 



where the subscripts x and 8 after a comma denote partial differentation 



with respect to the variable, i.e., 



_ 9v _ 9 w 



^'0 ='^'^'x0"i7^'^'^* 

 The quantities N^j^. and Ngy are the longitudinal and circumferential (hoop) 



thrusts per unit length, respectively, just prior to buckling. The longi- 

 tudinal thrust Nqjj. has the constant value (-pR/2) throughout. Kendrick 

 assumed the hoop thrust Ngy to have the constant value -pRhLf /(Af+Lfh) 



86 



