TORSION 119 



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



Mr 

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



p 

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



shaft of equal sectional area. 



Problem 122. An oak beam 6 in. square projects 4 ft. from a wall and is acted 

 upon at the free end by a twisting moment of 25,000 ft. Ib. How great is the angle 

 of twist? 



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

 equilateral triangle of side c, 



(/max = 20^, 



and the angle of twist per unit of length is 



e - M 

 **-T*5F 9 



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

 sional rigidity of a circular shaft of equal sectional area, 



102. 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 



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

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

 rigorously true for circular and elliptical shafts, and 

 St. Yfiumt has shown that it is approximately true 

 what.'\vr th>- form of cross section. 



Problem 123. Compare the angle of twist given by St. Venant's 

 il formula with the values given by the special formulas in 

 Articles 99, 100, and 101. 



Problem 124. Find the angle of twist in Problem 116. 



Problem 125. Find the angle of twist in Problem 117, and com- 

 pare it with the angle of twist for the solid shaft in Problem 124. 



103. Helical springs. The simplest form of a helical, 

 or spiral, spring is formed by wrapping a wire upon a 

 circular cylinder, the form of such a spring being that FlG - 87 



of a screw thread. Let r be the radius of the coil and a the radius 

 of the wire, and let the spring be either compressed or extended by 



