The last term in the second equation is based on the fact that the two 

 longitudinal sections bordering the element are not parallel. 



In order to be able to set up the equations for the w - direction, we 

 must know the value of the radius of curvature after the deformation. This is, 

 however, Q — O-— v/"— "^ ^ and since (conditions of equilibritim in the w - dl- 

 rection), (Translator's note: See Prescott's "Applied Elasticity", p.549i equa- 

 tion 17.101), "^— ^^' — ^Ni. ■ T^ _Q must hold, it follows, by the omission of 

 quantities of higher order: 



The two unknown shearing forces N, and N2 are determined with the help 

 of the equations showing that the summation of the moments about the u- and v- 

 directions are equal to zero. (See Love, p.536» eq,46) 



d-x. a. (9^ -2 adg> ()x ' ^10' 



Equations (8), (9), and (10), in combination with the elastic relations 

 of the previous section, give the complete statement for every problem dealing 

 with cylindrical shells. A particular solution corresponding to the symmetrical 

 compression through the external pressure £ is obtained with u = v = 0. It fol- 

 lows then that S = IT.|_ = N2 = Gj_ = G2 = H = 0, and from eq. (9): Tg = -ap. Fur- 



2 

 ther eq. (5) gives T^ =<?-T2 and eq. (1) C^ = - ^ = -~ , w= ^-2. . (Translator's 



note: The results are easily obtained since w is independent of z and 9> ). 



If we subtract this value of w from the actual displacement u, v, w, 



then the remaining part satisfies the same conditions, provided that the first 



term In the parenthesis in eq. (9) is dropped. (Translator's note: This means 



that the uniform radial compressive displacement is neglected in comparison with 



displacements due to buckling. That is, the displacements due to buckling are 



measxired from the original position of the neutral azis of the shell as Is shown 



in Figs. 4 to 6. Subtracting £_£ f rom w in (1) and substituting the resulting 



~ 



value of T? obtained from (3) in (9) gives equation (9') directly), in place of 

 equation {9) we have, therefore, 



JL^M'- ^^i - ^ (it- \ ^^^ "i (9') 



We can now, through the elimination of Nj^ and N2, derive the following 

 three oqulllbrixMi equations from equations (8), (9')» and (10): 



gi dlT _^ a -^1^— = (I) 



dx d^dx 



