244 PRACTICAL STRUCTURAL DESIGN 



Schwarz formula, an engineer of that name on the continent 

 of Europe having developed it independently of Rankine about 

 the same time. A number of other men proposed the same, or 

 a similar, form for the expression but Rankine and Schwarz ob- 

 tained the best publicity in advance of their colleagues. 



In the Rankine formula the constant " k " of the Gordon formula, 

 which was fixed by the shape and the material, becomes the con- 

 stant "a" which is fixed by the material alone. The constant 

 is modified by the method of supporting the ends of the column: 



Rankine formula for flat ends (fixed in direction) : 



l+o - )' 



For rounded ends (direction not fixed) multiply a by 4. 



For hinged ends (position fixed but direction not fixed) multi- 

 ply a by 2. 



For one end flat and the other round multiply a by 1.78. 



In the Rankine formula the compressive fiber stress was the 

 breaking strength and the reduced stress was the reduced break- 

 ing strength, which was divided by a factor of safety to obtain 

 the safe working stress. The same result is obtained by dividing 

 the breaking strength by the factor of safety. For example 



Rankine used c = 70,000 and a = onnnn - The stress / was di- 



vided by the factor of safety 5. To-day c = 14,000 and a = 



as before, but / is the safe unit stress without further operation. 



The values of the breaking stress and the empirical constants 

 to use in the Rankine formula were experimentally determined 

 by Christie and Hodgkinson many years ago as follows: 



Hard steel, c = 70,000 Ibs. per sq. in. a = ^ 



ZO 



Mild steel, c = 48,000 Ibs. per sq. in. a = 



30,000 

 Wrought iron, c = 36,000 Ibs. per sq. in. a = Q 



oD,UUU 



Cast iron, c = 80,000 Ibs. per sq. in. a = ^TTJO' 

 Timber, c = 7,200 Ibs. per sq. in. a = 



