o4 3 



_ w _ 9 w 



ax ax ae 



The assumed functions for the buckling displacements w and initial 

 out-of-roundness Wq for clamped-end cylinders are, respectively, 



r ZttxI 



^(x,e) = B sinnS 1- cos -f — 



e 1 Zttx I 



(x,6) = — sin n9 1- cos — - — 



(123) 



where e is the maximum amplitude of out-of-roundness and B is the buck- 

 ling coefficient. If the expressions (1Z3) happen to be an exact solution of 

 the problem, they will satisfy the differential equation of equilibrium. 

 Equation (119), exactly. However, as both w and Wq were chosen to satis- 

 fy the boundary conditions rather than the equilibrium equation, this, in 

 general, will not be the case. The resulting expression will be a function 

 of X and 6 which we shall denote by Q. In such a case, Galerkin's method 

 is used for determining the relations between the coefficients B and e; 

 this leads to the following condition: 



j j Q sinie 1- co£ 

 o o L 



Zttx 

 L 



Rdedx = ; i = 1,2,3, ... etc. (124) 



For i ijt n. Equation (124) will be found to be zero identically. For i = n 

 the following relation between B and e is obtained from Equation (124): 



B=T(nrrri^l (125) 



^IPcr-P, 



77 



