6 JOURNAL OF THE 



bars in trusses, etc., are added to make the paper suffi- 

 ciently complete in itself; especially as the aim is to make 

 the demonstrations as simple and elementary as possible. 



2. If we call a the length of a prismatic bar, tu its cross 

 section, e its modulus of elasticity and k its change of 

 length under the stress a, then, by the fundamental law of 

 elasticity, when the resultant stress j acts along the axis 

 and the limit of elasticity has not been exceeded, we have. 



(IK 



If the bar leno^thens or shortens the amount k bv the 



•;> 



action of a load P, acting in the direction of its axis, which 

 gradually increases from o to its greatest value P, the stress 

 likewise increases from o to its greatest value j, and at any 

 instant the stress is exactly equal to the load. If in any 

 small interval of time the average stress is s^ whilst the 

 length changes an amount d k, due to a slight increment 

 of load, the work done is s. d k; therefore as the load 

 changes from o to P, the total work done is the limit of the 

 sum, 



1 (s. d k) = 



between the extreme values k ^ o and k = k, correspond- 

 ing to s ^ o, and s = s. 



. '. The total 7vork of deforniation for ^gradually applied 

 load is, 



I .^ I k^ I T 



— I kdk= = _cs2 = — sk.... (2); 



CO c 2 2 2 



or since load P = s, work = \ load X change of length. 



