278 



DK E. U. COKER ON 



From the mathematical point of view, Thomson's conclusions may be arrived at as 

 follows : — 



If a specimen be subjected to stress sufficient to cause a uniform shear throughout, 

 and then be released, we have a new distribution of shear throughout the section, 

 which may be expressed by 



Shear = q — o/r 

 where q„ = original shear at the external radius. 

 r = any radius. 

 a = a constant. 

 Since the bar is in equilibrium, we must have 



/: 



giving 



" (//„ - ar)2Trr 2 dr = O 



4 j. 



3 r»' 



Thus the shear in the bar is given by the expression 



4 r 



The distribution m.Sij be shown graphically, as in fig. 21, 

 by a line AB, where OB = q and AC = 



3 



This line evidently crosses the axis OR at the distance 

 3 

 OE=-r, and gives a point on the circle of no stress. 



Clearly, if no change has taken place in the limits of 

 elasticity, the maximum shear is q at the centre, and 

 evidently the bar will now stand a torque given by the 

 expression 



Fig. 21. 



T = 



4 So . /-, 4 r, 

 o — r + q.\ 1 — o " 

 .3 r„ x "\ 3 r 



2-jrrWr = g Trqji ; 



while in the opposite direction the torque will be given by the expression 



T= \ r " 



v>, 



q„ 



1 - 



4 r\ 



2-7rr 2 dr = 5 7rq ri . 



3 r '-*°V 3 rj_ 



It remains to be seen if the assumptions are justified. 



In order to examine this point, a wrought-iron specimen was taken, having a length 

 under test of 4 - 00 inches; diameter 0*634; calibration value 1 min. = 12*76 divisions 

 of the scale. 



The specimen was set in the machine so that torque could be applied in either 

 direction, and observations were made of the strains for loads, which in turn caused 

 permanent set in both the positive and negative directions. 



These readings are plotted in fig. 22, and from an inspection of this it is ap- 

 parent that the stress-strain curve was approximately linear before the yield-point 



