190 



COLUMN FORMULAS. 



Art. 113. 



isfactory value can be assigned to the eccentricity. If sec. 61 is 

 developed into a series, and the higher powers omitted, equation 

 (52) becomes: 



When e is very small when care has ben taken to have the 

 load concentric and the column straight , . 2 will be less than 



evi P 

 e and may be neglected. Since e is unknown, ^ -r-^ may be set 



equal to (since it will be a small fraction) which is to be de- 



termined by experiment. We have then the following formula 

 which, in this country is called Rankine's Formula. 





(61) 



This equation is best solved by trial, since r depends upon 

 the section sought, but the solution is very simple compared with 

 that of equation (52). 



Equation (61) represents a curve which is laid in the middle 

 of a field of results, plotted from experiments made upon columns 



having various ratios of . The value of a so determined would 

 not, of course, apply beyond the limits of used in the experi- 

 ments, or to any other end conditions or kind of material ; but it is 

 assumed to apply to any form of cross section. Even within these 

 limitations, a is evidently not a constant, but the formula can be 

 made to represent the experiments very closely between certain 



limits of - ; this is sometimes better accomplished by putting 



for 5 max , a value different from its value in tension. It is not per- 

 missible in all cases to omit the higher powers in the development 

 of the series for sec 61. Rankine's Formula, therefore, is purely 

 empirical. It was originally deduced by Navier from equation 

 (51) in a manner which gave a much simpler value of a than 

 that given above, and which shows more clearly that a cannot be 

 a constant, but it is none the less indeterminate theoretically. 



