110 KESISTANCE OF MATERIALS 



Note that Rankine's formula gives the principal normal stresses, that 

 is, tension or compression, whereas Guest's formula gives shear. Since 

 the ultimate strength in tension or compression is usually different 

 from that in shear,* in designing circular shafts carrying combined 

 stress both formulas should be tried with the same working stress (or 

 factor of safety), and the one used which gives the larger dimensions. 

 65. Resilience of circular shafts. In article 7 the resilience of a 

 body was defined as the internal work of deformation. For a solid 

 circular shaft this internal work is 



where M t is the external twisting moment and 6 is the angle of twist. 



From equation (137), = = 



Gr Ga 



and from equation (138), 



Therefore the total resilience of the shaft is 



(153) 

 and consequently the mean resilience per unit of volume is 



;-?- 



66. Non-circular shafts. The above investigation of the distribu- 

 tion and intensity of torsional stress applies only to shafts of circular 

 section. For other forms of cross section the results are entirely 

 different, each form having its own peculiar distribution of stress. 



For any form of cross section whatever, the stress at the boundary 

 must be tangential, for if the stress is not tangential, it can be 

 resolved into two components, one tangential and the other normal 

 to the boundary ; but a normal component would necessitate forces 

 parallel to the axis of the shaft, which are excluded by hypothesis. 



Since the stress at the boundary must be tangential, the circular 

 section is the only one for which the stress is perpendicular to a 

 radius vector. Therefore the circular section is the only one *to 



* The shearing strength of ductile materials, both at the elastic limit and at the ulti- 

 mate stress, is about four fifths of their tensile strength at these points. 



