96 RESISTANCE OF MATERIALS 



Gordon, Rankine, Navier, and Schwarz.* Among German writers 

 it is known as Schwarz's formula, but in English and American 

 textbooks it is called Rankine's formula. 



For I = 0, P = gA, and, since short blocks fail by crushing, g is 

 therefore the ultimate compressive strength of the material. 



For different methods of end support Rankine's formula takes 

 the following forms: 



(125) ^ = ^-/7\* Ftat ends 



^ _i_ -P / _ | (fixed in direction) 



w 



P Q 



(126) = y 2 Round ends 



* 1 _|_ 4 f I _ \ (direction not fixed) 



J \t) 



P Q 



(127) = y 2 Hinged ends 



1 4- 2 f ( 1 (position fixed, but not 



1/1 direction) 



P # 



(128) = 77-3 One end flat and the 



1 _j_ 1.78/ f\ other round 



56. Values of the empirical constants in Rankine's formula. The 



values of the empirical constants, / and g, in Rankine's formula 

 have been experimentally determined by Hodgkinson and Christie, 

 with the following results : 



For hard steel, g = 69,000 lb./in. 2 , /= 



For mild steel, g = 48,000 lb./in. 2 , / = 



For wrought iron, g = 36,000 lb./in. 2 , /= 

 For cast iron, g = 80,000 lb./in. 2 , / = 



20000 



1 

 30000 



1 

 36000 



1 

 6400* 



For timber, ff = 7,200 lb./in. 2 , / 



* 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 center of the column is 



given by the expression D = MT- where J is the length of the column, 6 its least width, 

 and M an empirical constant. 



