TORSION 113 



From Hooke's law (Article 33), %. = G. Hence 



<P 



(54) q = G+ 



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

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

 and attaining its maximum value at the circumference. 



If q' denotes the intensity of the shear at the circumference and 

 a denotes the radius of the shaft, then the shear q at a distance r 

 from the center is given by the formula 



q'r 

 =a 



Let M denote the external twisting moment. Then, since M must 

 be equal to the internal moment of resistance, 



M= C fJ rdF = C 7 *dF 



win-re I p is the polar moment of inertia of the section. 



For a solid circular shaft I p = - - > and consequently 



2 



, Ma 2 .I/ 



'' : 



For a hollow circular shaft of external radius a and internal radius 

 b, I p = > and hence 



< 56 > *-*&=?> 



94. Angle of twist in circular shafts. From equation (54), 



~ Gr~~Ga 

 Therefore, for a solid circular shaft, from equation (55), 



