ELISHA MITCHELL SCIENTIFIC SOCIETY. 9 



caused by the stress 5 due to all the external forces such as 

 P (the force unity being omitted), and that this elongation 

 alone causes the displacement /\ p^ of force unity in the 

 direction of that force. In applying the principle of vir- 

 tual velocities we have the right to suppose any displace- 

 ment A 1 we choose, and for convenience we take that the 

 bar actually sustains when the truss is fully loaded and not 

 what it would sustain from the force unity. 



Now assuming ic to be tension, the displacements of the 

 ends of the bar are in the opposite directions to the forces 

 acting, so that the virtual velocity is negative. We shall 

 assume the displacement A p^ to be in the direction of the 

 force unity until otherwise ascertained. 



We have now by the principle of virtual velocities, 

 I. A p^ — u. A 1 = o; 

 a s 



.-. A p' = u . 



e w 



If u or .V are compressive, it is evident that they must 

 have the minus sign in the above equation. Should A p^ 

 thus become minus in any case, the displacement will be 

 contrary to the direction of the supposed force unity. 



Continuing thus to find the displacement of force i, due 

 to the change of length of each bar in turn, the other bars 

 remaining unchanged, we have for the total displacement 

 of the force unity, acting in the direction of external force 

 P, the formula, 



\ e w/ 



A p = -i u I . . . . (4), 



the sum extending to all the bars of the truss. 



But since this displacement is that caused by the actual 

 stresses in all the bars due to the original external forces, 

 it must equal the actual displacement of force P along its 

 direction or the deflection of the truss in the direction of 

 force P. 



