COLUMNS AND STRUTS 95 



of a cross section, the breaking load for a very short column is 

 P=pA* 



For columns of ordinary length, therefore, the load P must lie 

 somewhere between pA and the value given by Euler's formula. 

 Consequently, to obtain a general formula which shall apply to 

 columns of any length, it is only necessary to express a continuous 



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relation between pA and . Such a relation is furnished by the 

 equation 



(123) P = 



1+pA 



For when I = 0, P =pA, and when I becomes very large, P approaches- 



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the value - Moreover, for intermediate values of I this formula 



gives values of P considerably less than those given by Euler's 

 formula, thus agreeing more closely with experiment. 



55. Rankine's formula. Although the above modification of 

 Euler's formula is an improvement on the latter, it does not yet 

 agree closely enough with experiment to be entirely satisfactory. 

 The reason for the discrepancy between the results given by this 

 formula and those obtained from actual tests is that the assumptions 

 upon which the formula is based, namely, that the column is perfectly 

 straight, the material perfectly homogeneous, and the load applied 

 exactly at the centers of gravity of the ends, are never actually 

 realized in practice. 



To obtain a more accurate formula, two empirical constants will 

 be introduced into equation (123). Thus, for fixed ends, let 



(124) 



where /and g are arbitrary constants to be determined by experi- 

 ment, and t is the least radius of gyration of a cross section of the 

 column. This formula has been obtained in different ways by 



* As Euler's formula is based upon the assumption that the column is of sufficient 

 length to buckle sideways, it is evident a priori that it cannot be applied to very short 

 columns, in which this tendency is practically zero. Thus, in formula (1 1 8) , as I approaches 

 zero P approaches infinity, which of course is inadmissible. 



