where a , a , a .... etc. are rather complicated expressions of the 

 11 12 13 ^ ^ 



geometry and loading; see Equations (27) in Reference 60. From 

 Equations (173), the displacement coefficients A, B, and C may be deter- 

 mined in terms of the out-of-roundness amplitude Cq. The case Cq = 

 leads us back to the eigenvalue problem of the elastic general instability. 

 Once the coefficients A, B, and C are determined in terms of Cq, the 

 elastic stresses due to imperfect circularity, at any point on the periphery 

 of the frame flange, can be found in terms of the amplitude Cq from the 



equation 



-£ = -'= 



,- av, 1 [ a w av\ ^f 



(174) 



where the displacements u, v, and w are given by Equation (172); Equation 

 (174) is derived in Reference 60. The first term in the brackets of 

 Equation (174) represents a direct-stress component whereas the second 

 term represents a bending component; both components are due to the 

 asymmetric bending action as a consequence of the out-of-roundness. To 

 obtain the total stress in the flange of a ring frame, the axisymmetric mem- 

 brane component due to the radial load Q* from Equation (21) must be added 

 to that of Equation (174); this latter stress is given by 



%f = r~ (175) 



^' (A ^^ +bh)(R +d + ^) 

 eff 2 



where d is the depth of the circular ring frame. Equation (175) is valid 

 for external frames; for internal frames, the factor (R - d- — ) must be 

 used in the denominator (see Figure 11). 



Ill 



