the pressure p* 2h which acts on the surface of the cross-section of the element, 

 produces a resultant in the w direction (radially inward, hence negative) equal to 



-P' 2h P (2) 



This expression, with opposite sign, must therefore be added on the right 

 side of equation (9) in Z, if we would take into account a general end loading. 

 Substituting the value of Eq. (1) in (2) we have: 



-oa a -1^- 



(3) 



3 2 w 

 We must, therefore, instead of - -g—g- in the parenthesis on the right hand side 



of Eq. (9) in Z, write 



7) a w _ a 2 ^ 2 w 

 IJf 5 " 2 llx* 



The same remarks apply to Eq. (9 1 ) and (III) in Z. The end loading has no further 

 influence on the expressions for the determination of the buckling pressure, as 

 long as we remain within the limits of the simplifying assumptions introduced in Z 

 or presupposed. 



It is not difficult to follow the influence of the supplementary term, 

 equation (3), through to the final formula in Z. 



From Eq. (11) of Z and the abbreviation or given in Eq. (12) of Z, we have 



a2 tf - = -* 2w ' * - -r 1 (4) 



(1 = effective length or frame spacing; see below, Section 2). 

 Therefore, 



y (1 - n 2 - <Z 2 /2) 

 is to be substituted on the right hand side of Eq. (Ill') of Z in place of y(1 - n 2 ): 

 n represents the number of lobes. This same substitution is valid for the last 

 member in the determinant (16) of Z. Since now y is the only quantity in the final 

 equation that contains p, and, therefore, is directly proportional to p, it follows 

 that: The buckling pressure of a tube, including the effect of end load, may be 

 obtained by multiplying the pressure determined from Z by the factor 



n 2 - 1 , e . 



— m (5) 



This statement applies primarily to the complete result of the calculation 

 carried but in Z, Eq. (A) or (A*), but it may be directly applied also to the 

 simplified expression (B), since the assumptions that led to B smallness of 



