22 TORSION. 



on the shaft as equal to the sum of the moments of the 

 internal forces, or 



= 2?r / r v 

 v J x 3 



dx 



If we use the diameter d instead of the radius ?*, the 

 formula becomes 



T =!^ / & ..... (IX.) 



Where the shaft is hollow, with the two diameters D and d, 

 we have 



T jr , 

 " 7 



4 ..... (X.) 



16 D 



So far we have taken no account of strains. When the 

 short cylinder in the figure is strained by an external 

 twisting moment T, the line A B becomes deflected into 

 a position A C. Call the angle BAG, #. Now 



Let the length of the cylinder be I, and the angle 

 COB, <. Then, for small deflections 



and 



TJ /~1 



4> = , therefore 

 r 



=!*= 



i \j 

 wherefore, 



I f 



r' G 

 From (VIII.) < = - -^ (XL) 



TT 7 (jr 



writing this otherwise, we get 



m .G.^) 4 



"2" I 

 Another useful form is 



G = 2 - L T , (XIII.) 



7T. (p. 7* . 



= 32. ?. T 



7T. <f>. d . 



