term Fourier series, the convergence of which is dependent upon the shell- 

 flexibility parameter 6 or p , to express the deformations permitted a 

 closer approximation to the true state of stress prior to buckling. By 

 including the torsional as well as the bending energies of the ring frames, 

 Reynolds was able to consider all degrees of elastic support afforded the 

 shell by the ring frames, ranging between and including the two extreme 

 cases of simple supports and clamped conditions. 



The energy integrals and some of the intermediate mathematical 

 operations used by Reynolds are rather lengthy and cumbersome so that 

 these details will not be considered here. However, it is of interest to 

 give the final equation from which the critical elastic panel-buckling 

 pressures can be computed. Thus, for the most general case of interest 

 to the designer, that of ring frames possessing finite bending and torsional 

 stiffnesses, Reynolds derived the following buckling critericm: 



1 + S 



H in 

 1 1 



^ °i 



1 + 2 



^i'-i 



i °i 



G U H X 

 i i i i o 



- T — — y = 



. D . D 



1 1 1 i 



(62) 



where i = 1, 3, 5, . . . and, N is the summation index specifying the 

 number of terms to be taken in the Fourier expansions to get varying 

 degrees of numerical convergence. The other quantities appearing in 

 Equation (62) are defined as follows: 

 2 





vl\ 



(63) 



54 



