SHEARING. 19 



When the stresses, S, produce distortion through a small 

 angle 9, 6 being measured in circular measure, we have the 

 relation 



S 

 -p- = 0. This is for shearing. 



The equivalent stresses in tension f t and compression 

 / c may be considered as producing an elongation of the 

 diagonal A C, causing it to assume a length A c. This elonga- 

 tion is due to the tensile stress acting along and parallel 

 to the diagonal and also to the compressive stress f e , acting 

 at right angles to this. The combined effect of these is to 

 produce a strain equal to A c A C. Thus, we have the 

 relation 



AC-AC _/, /, _i_ 



AC " E h E ' m 

 where m is Poisson's Ratio, or 



Ac-AC_/ / JL\ 

 AC E V m/ 



ft being = f e = f 

 It will be obvious that, for any distortion, the angle 



E A b = 2 (angle C A c) ; or, C A c is-ift 



z 



If the extension A c A C be called q, from the figure 

 it will be seen that 



q = Ac - AC 



= 2. A B. cos(4 ! } - 2. A B. cos^- 

 \ 4 z/ 4 



If is very small, this expression reduces to 



q = A B.-^= 

 V2 



and AC = A/2~AB; 



So that 



Ac- AC 



. 

 becomes -^ 



But 



IjT 



Therefore we have, finally, 



SL = . /A AN 



G 2 2 E V m) 



( S =/,=/,= ) 



