Lb 



When q is given the value ip: — , the shape of the radial deflection w 



between bulkheads is a half-sine wave. Since there are seven arbitrary 

 coefficients in the buckling deformations, Equation (160), the system 

 possesses, in a sense, seven degrees of freedom which allow independent 

 deformations between the light frames, the heavy frames, and the ends of 

 the cylinder; the three degrees associated with the w component of dis- 

 placement are shown in Figure 18. 



Reynolds pointed out that the generality of the analysis could be 

 improved by permitting one additional degree of freedom. This was done 

 by adding to the axial displacement u(x,e) a second component varying 

 periodically between adjacent heavy frames, i.e., Reynolds suggested the 

 following function instead of that given in Equation (160): 



u(x,e) = AjU^(x) + ^^ A3 — |sin— I cosne (164) 



This is discussed further in Reference 58. 



The method of solution then goes along the following lines: the 

 assumed buckling displacements, Equations (160) and (164), are substituted 

 into the integrals for the shell and ring energies and for the work done, and 

 thus the total potential, Equation (159), is determined. The condition of 

 minimum potential energy, i.e., 6Vrp = (see Equation (145), for example), 

 leads to a system of eight linear homogeneous algebraic equations for the 

 eight coefficients Aj , A3, B^, B^, B3, Cj, Cz, and C3. The 8X8 stability 

 determinant formed by the coefficients of the A's, B's, and C's when 



103 



