TORSION 



111 



which the above development applies, and consequently is the only 

 form of cross section for which Bernoulli's assumption holds true. 

 That is to say, the circular section is the only form of cross section 

 which remains plane under a torsional strain. 



The subject of the distribution of stress in non-circular shafts has 

 been investigated by St. Venant, and the results of his investigations 

 are summarized below (articles 67~70). 



67. Elliptical shaft. For a shaft the cross section of which is an 

 ellipse of semi-axes a and 6, the maximum stress occurs at the ends 

 of the minor axis instead of at the ends of the major axis, as might 

 be expected. The unit stress at the ends of the minor axis is given 

 by the formula 



tfmax = 



(154) 



and the angle of twist per unit of length is 



(155) 



The total angle of twist for an elliptical shaft of length I is therefore 

 (156) 6 = 0J = 33 



68. Rectangular and square shafts. For a shaft of rectangular 

 cross section the maximum stress occurs at the centers of the longer 

 sides, its value at these points being 



.68 + .45 - 



(157) 



hb vh + 



in which h is the longer and b the shorter side of the rectangle. The 

 angle of twist per unit of length is, in this case, 



(158) 



\ = 3.57 



For a square shaft of side b these formulas become 



(159) 

 and 

 (160) 



= 4.8 



X = 7.14 



Gb 4 



