TWISTING MOMENT 318 



AC 



Now Shear strain = -r^ 



AD 



- tan <p 



-= 9, if 9 is a small angle. 



Thus 9 radians measures the shear strain at the surface of the 

 cylinder. 

 Also AC = fcp = r0 



r6 

 and <p = 







But shear stress = N x shear strain where N is the modulus of 

 rigidity. 

 Then f t = N^ 



N0 



where/, is the shear stress at the surface of the cylinder. 



Let q be the intensity of the shear stress at a point in the 

 cylinder whose radial distance is x. 



NO 

 Then q = j- x 



= -f 



r Js 



Let this shear stress take place over an elementary ring of 

 width Bx, situated at a radial distance x. 



Area of elementary ring = 2icx x 

 Total shearing force = q x area 



V 



The moment of this force about the axis of the cylinder 



27C - cv 



27T f r 



Hence T = twisting moment = f t \ x 3 dx 



f Jo 



7T 



t where d is the diameter. 



