CALCULATION OF THE REACTIONS 129 



movement of the truss A' at the hinge Z/ will then be due to the 

 change in length, 8, of the member i-y. 



Let h 1 be the horizontal reaction necessary to bring L 1 back to 

 its original position, and let U h 1 be the stress in the member i-y due 

 to the horizontal thrust k 1 . Now the internal work, y 2 8 h 1 U, in short- 

 ening the member 1-3; to its original length will be equal to the external 

 work, y 2 h 1 A', required to bring the hinge LJ- back to its original 

 position, 



y 2 h> A' = % 8fr U 

 and A' = 5 U (65) 



PL 



but 5 = , where P is the unit stress in the member i-y due to the 



external load W, L is the length of the member i-y, and H is the mod- 

 ulus of elasticity of the material of which the member is composed. 

 Substituting this value of 8 in (65) we have 



(66) 



where U is the stress in i-y due to a load ^p = unity at L\ 



Now if each one of the remaining members of the arch is assumed 

 to be distorted in turn, the others meanwhile remaining rigid, the dis- 

 tortion in each case at L\ will be represented by the general equation 

 (66) and the total deformation, A , at L 1 will be 



(67) 



Let P 1 h 1 be the unit stress in the member i-y due to a horizontal 

 thrust h 1 , then by the same reasoning 



A' = 8 U (65a) 



P l h> L 



but 5 = -g 



& P* UL 

 and A' - ^ 



and the total deformation, A, will be 



