112 RESISTANCE OF MATERIALS 



The value of q for a square shaft found from this equation is 



Mr 

 about 15 per cent greater than if the formula q = were used, and 



P 

 the torsional rigidity is about .88 of the torsional rigidity of a 



circular shaft of equal sectional area. 



69. Triangular shafts. For a shaft whose cross section is an 

 equilateral triangle of side c, 



(161) 



and the angle of twist per unit of length is 

 (162) 



The torsional rigidity of a triangular shaft is therefore .73 of the 

 torsional rigidity of a circular shaft of equal sectional area. 



70. Angle of twist for shafts in general. The formula for the 

 angle of twist per unit of length for circular and elliptical shafts 

 can be written 



(163) B, : 



in which I p is the polar moment of inertia of a cross section about 

 its center, and A is the area of the cross section. This formula is 

 rigorously true for circular and elliptical shafts, and St. Venant has 

 shown that it is approximately true whatever the form of cross 

 section. 



APPLICATIONS 



186. A steel wire 20 in. long and .182 in. in diameter is twisted by a moment 

 of 20 in.-lb. The angle of twist is then measured and found to be = 18 31'. What 

 is the value of G determined from this experiment ? 



Solution. From equation (140), article 62, 



G = 



where, in the present case, using pound and inch units, M t = 20, I = 20, D = 182, 

 and 6 = 18 31' = .3232 radians. Substituting these numerical values, the result is 



G = 11,490,000 lb./in.2 



187. A steel shaft Sin. in diameter is driven by a crank of 12-in. throw, the 

 maximum thrust on the crank being 10 tons. If the outer edge of the shaft-bearing 

 is 11 in. from the center of the crank pin, what is the stress in the shaft at this point ? 



