118 STEENGTH OF MATERIALS 



cross section for which Bernoulli's assumption holds true. That 

 say, the circular section is the only form of cross section which rem 

 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 in\vsti^aiins 

 are summarized below (Articles 99-102 inclusive. 



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

 ellipse of semi-axes a and b, the maximum stress occurs at tin- <-nls 

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

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

 by the formula 



and the angle of twist per unit of length is 



.!/( + fr 



01 = 



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

 e _ ei ^ J/fr' + ft 8 )/ 



Problem 121. The semi-axes of the cross section of an elliptical shaft are 8 in. 

 and 5 in. respectively. What is the diameter of a circular shaft of equal strength ? 



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



cross section the maximum stress occurs at the centers of the In: 

 sides, its value at these points being 



(61) 



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, 



For a square shaft of side b these formulas become 



(62) 

 and 



Gb* 



