Mr. .!.('. M -Gunnel. 



[Mar. li>, 



( htstic bar, supported at either end and loaded in the middle, had 

 been fully worked out. But this does not. appear to be Hhe case.y 

 The ordinary elementary treatment makes the gigantic assumption ., 

 that plane cross-sections of the unbent bar remain plane, and that thet 

 lateral contraction or expansion of elementary strips parallel to the 

 ength of the bar under longitudinal pulls or thrusts are the same ns in 2 

 free space. It does not consider any shearing stresses or strains. It is 

 true that Rankine ('Applied Mechanics,' p. 338), assuming Hie 

 results of this method, proceeds to find an expression for the shearing 

 stress. He makes it proportional to a 9 a: 2 , where the origin is at 

 the centre of the bar, the axis of x is drawn upwards, and 2a is 

 the depth of the bar. But this expression is inconsistent with the 

 general equations of an elastic solid. St. Tenant's solution of thej 

 bending of a bar, given in Thomson and Tait's ' Natural Philosophy,*' 

 postulates equal and opposite couples applied at the two ends, so that < 

 the bending moment is uniform throughout. The importance of the 

 absence of this uniformity is not trifling but fundamental, for in onrl 

 case everything depends on the shears, and in St. Venant's solution! 

 there are no shears. 



I fancy that I see my way to obtaining the complete solution in 

 the form of infinite series. But, since it ceases to be applicable the} 

 moment plastic strains take place, it would only enable us to deter- 1 

 mine the inUial stresses, and this would hardly justify the insertion i 

 here of such a long investigation. 



The following simple but imperfect treatment must suffice. Let 

 ns first define the coefficient of plasticity. Take a rectangular element* 

 with two faces normal to the optic axis, and let these faces be sub- 

 jected to a tangential force U per unit of area in opposite directions, 

 parallel to another pair of faces. 



Fio. 8. 



U 



Then if the rate of growth of two of the angles, or rate of dimi- 

 nution of the other two be denoted by dx/dt, the coefficient of plas- 

 ticity p may be defind by the equation 



