TORSION. 21 



The two planes are rotated, as shown by the arrows 

 (Fig. 8), causing a distortion. A line A B originally parallel 

 to the axis is brought to A C, the angle BAG being called 6. 



This is similar to the angle of distortion of the cube which 

 we have just considered. The radius B is turned through 

 an angle B C = </>. 



Consider a small ring of the circle at a radius x and 

 having a width 8x. The whole area of such a strip will be 



= 2 TT x 8x 



The stress on the metal due to the rotation will be a shear- 

 ing stress on the section of the shaft perpendicular to the 

 axis. The intensity of this stress, unlike the case of simple 

 shearing where it is constant over the whole area, will vary 

 from centre to circumference, and will be proportional 

 to its distance from the axis. If / be the stress at 

 the surface of the shaft and s be the stress at any other 

 radius x, f will be the maximum stress corresponding to 

 the maximum strain, and the intensity of the stress s at a; 

 will be 



, x 



s = f- 

 - r 



r, being the radius of the shaft. Therefore, the total force 

 exerted upon the ring in question will be 



s. 2 TT x. 8 x. ; 

 and its moment about the axis 



s. 2. TT. x. &x. x, or 



2. -Ky? 8 x 

 r 



Integrating this, we have the . external twisting moment 



