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1 . DERIVATION of the FUNDAMENTAL FORMULA. 



Let us consider a thin-walled, circular, cylindrical, hollow body, closed 

 at the ends by flat heads, and subjected to the pressure p on its shell and to 



the compressive force P* on its 

 heads; Fig. 1. 



Let the axial pressure on a unit 

 ,' cross-section of the circular ring hav- 

 — ing a diameter 2a and thickness 2h have 

 a value p*. If the pressure on the 

 heads is due to the fact that the end 

 surfaces are subjected to the external 

 pressure p, it follows from: 



P+-= 7Ta a p = IT 2a 2h p' 



that p' = p a/4h (1) 



In order to take into consideration the presence of an end pressure p 1 the condi- 

 tions of equilibrium of Z must be supplemented. For those equations are incom- 

 plete inasmuch as only terms of the first order in stress and deformation magni- 

 tudes are retained in them. Products of stresses and deformations of the second 

 order of magnitude are omitted, with the exception of the products of p by w and 

 2jtt» (Z, Eq. (9)) since p, unlike all other stresses, is not of the order of 

 magnitude of the deformations accompanying buckling but possesses a finite value 

 independent of them. We must now, in an analogous manner, include also products 

 of p* with deformation magnitudes in our equations. 



FIG. 1 



FIG. 2 



Fig. 2 shows an element of the shell in a longitudinal plane through the 

 axis of the tube, in the deformed condition. The generatrix, originally rectilin- 

 ear and parallel to the x-axis, has acquired a curvature, which, except for terms 

 of higher order, is measured by 



3 8 w 

 ax z 



This expression, in the case of Fig. 2, is negative. It is seen that 



