TORSION 



113 



Solution. Referring to Fig. 88, the 

 imensions in the present case are 



^ = 12 in., eZ 2 = 11 in. 

 iquently, 



M t = 10 2000 12 = 240,000 in.-lb. 

 M b = 10 2000 . 11 = 220,000 in.-lb. 



Therefore, from equation (150), article 64, 

 ,p max = -15- (220, 000 + V220,000 2 + 240,000 2 ) 



= 22,200 lb./in. 2 , 

 and similarly, from equation (151), 



Wo 



FIG. 88 



= ( V220,000 2 + 240,000 a ) = 13,275 lb./in. 2 



188. If P and Q denote the unit stresses at the elastic limits of a material in 



p 

 tension and shear respectively, show that when < 1 the material will fail in ten- 



P 



sion, whereas when > 1 it will fail in shear, when subjected to combined bending 



TM 



and torsion, irrespective of the relative values of the bending and twisting moments. 

 Solution. Combining Rankine's and Guest's formulas, we have 



Consequently, if the bending moment is zero, p' = q', or = 1, whereas if it is not 



zero, p' > q'. Similarly, if the twisting moment is zero, f = 2. 



Now let F t and F 8 denote the factors of safety in tension and shear respectively. 

 Then 



F 8 Q Qp' 



' 

 Since p'isg', the fraction ^1. Consequently, if <1 also, then J<l;that 



p' Q f g 



is, F t < F,, and the material is weaker in tension than in shear. The second part 

 of the theorem is proved in a similar manner. 



For a complete discussion of this question see article by A. L. Jenkins, En- 

 gineering (London, November 12, 1909), pp. 637-639. 



189. Three pulleys of radii 8, 4, and 6 in. respectively are keyed on a shaft as 

 shown in Fig. 89. Pulley No. 1 is the driving pulley and transmits 30 H.P. to the 

 shaft, of which amount 10 H.P. is taken off from pulley No. 2 and the remaining 

 20 H.P. from pulley No. 3, The speed is 50 R.P.M., the belts are all parallel, and 

 the tension in the slack side of each belt is assumed to be one half the tension in the 

 tight side. Find the required size of the shaft for a working stress of 12,000 lb./in. 2 

 >in tension and 9000 lb./in. 2 in shear. 



