COLUMNS AND STRUTS 



97 



These constants were determined by experiments upon columns for 

 which 20 < - < 200, and therefore can only be relied upon to 



v 



furnish accurate results when the dimensions of the column lie 

 within these limits. 



As a factor of safety to be used in applying the formula, Rankine 

 recommended 10 for timber, 4 for iron under dead load, and 5 for 

 iron under moving load. 



57. Johnson's parabolic formula. From the manner in which 

 equation (123) was obtained and afterwards modified by the intro- 

 duction of the empirical constants / and #, it is clear that Rankine's 

 formula satisfies the requirements for very long or very short col- 

 umns, while for those of intermediate length it gives the average 

 values of experimental results. A simple formula which fulfills 

 these same requirements has been given by Professor J. B. Johnson, 

 and is called Johnson's parabolic formula. 



If equation (124) is written 



- = ff 



A 



t) 



and then y is written for JP, and x for -, Rankine's formula becomes 



For this cubic equation Johnson substituted the parabola 



(129) 



y = 



in which x and y have the same meaning as above, and 8 and e are 

 empirical constants. The constants 8 and e are then so chosen that 

 the vertex of this parabola is at the elastic limit of the material 

 on the axis of loads (or F-axis), and the parabola is also tangent 

 to Euler's curve. In this way the formula is made to satisfy the 

 theoretical requirements for very long or very short columns, and 

 for those of intermediate length it is found to agree closely with 

 experiment. 



