TORSION 



107 



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' 



Now if q denotes the intensity of the shear on any element of 

 area A^4, the total force acting 011 this element is <?A^4, and its mo- 

 ment with respect to the center is qAAr. Therefore the total internal 

 moment of resistance is ^q&Ar, where the summation extends over 

 the entire cross section ; and since this must be equal to the exter- 

 nal twisting moment M# we have 



(J T 



Inserting for q its value in terms of the radius, q = - , this becomes 



or, since by definition Vr 2 AJ. = I p , the polar moment of inertia of 

 the cross section, 



For a solid circular shaft of diameter Z>, I p = -^- and a = ; 



M t a 16 M t 



consequently, 

 (138) 



For a hollow circular shaft of external diameter D and internal 



7T D 



diameter d, I p = ^ (D 4 d 4 ) and a = ; hence 



32 



(139) 



62. Angle of twist in circular shafts. From equation (137), 



Gr Ga 



Therefore, for a solid circular shaft, from equation (138), 



<"> *=Sr 



