SECTION IX 



TORSION 



61. Maximum stress in circular shafts. When a uniform circular 

 shaft, such as is shown in Fig. 87, is twisted by the application of 

 moments of opposite signs to its ends, every straight line AB paral- 

 lel to its axis is deformed into part of a helix, or screw thread, AC. 

 The strain in this case is one of pure shear and is called torsion. 



h 



The angle < is called the angle of shear and is proportional to the 

 radius BD of the shaft. The angle 6 is called the angle of twist 

 and is proportional to the length AB of the shaft. 



Consider a section of length A# cut from a circular shaft by 

 planes perpendicular to its axis (Fig. 87). Let A0 denote the angle 

 of twist for this section. Then, since the angle of twist is propor- 

 tional to the length of the shaft, A0 : 6 = A# : I ; whence 



A<9=0 



I 



Also, if </> and A0 are expressed in circular measure, 

 BC=<j>-AB = <j>kx, and BC = A0 . BD = 



Therefore <f> = = r From Hooke's law, = G. Hence 



(137) 



Therefore q is proportional to r; that is to say, the unit shear is 

 proportional to its distance from the center, being zero at the 

 center and attaining its maximum value at the circumference. 



106 



