104 STRENGTH OF MATERIALS 



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

 somewhere between pF and the value given by Euler's f..nimla. 

 Consequently, to obtain a general formula which shall app! . 

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

 relation between ^F and 77-' Sucn a relation is furnished by the 



equation 



pF 

 (51) P= W 



1+pF 



For when / = 0, P = pF, and when / becomes very large P approaches 



the value ^- Moreover, for intermediate values of / this fm-mula 



I 



gives values of P considerably less than given 1>\ Kuler's formula, thus 

 agreeing more closely with experiment. 



86. Rankine's formula. Although the above moditiratiun of 

 Euler's formula is an improvement <n the latter, it does not 

 agree closely enough with experiment to be entirely satisfactory. 

 The reason for the discrepancy between the results given hy this 

 formula and those obtained from actual tests is that tlu assumptions 

 upon which the formula is based, namely, that the colum < tly 



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 (51). Thus, for fixed ends, let 



< 



'*' 



(!) 



where /and g are arbitrary constants to be determined by experiment. 

 and t is the least radius of gyration of a cross section of the column. 

 This formula has been obtained in different ways by Gordon. I.'an- 

 kine, Navier, and Schwarz.* Among German writers it is knov 



*Rankine's formula can be derived independently of Euler's formula either by 

 assuming that the elastic curve assumed by the center line of the column is a sinusoid, 

 or by assuming that the maximum lateral deflection D at the cent.T ..f th- olumn is 

 given by the expression D = /i^, where I is the length of the column, 6 its least width, 

 and n an empirical constant. " 



