TESTING OF STRUTS. 283 



For very long pillars, where the ratio of length to 

 diameter is greater than 150 to 1, the true relation is 

 given by Euler's Formula*, which is as follows : 



* PROOF OP EULER'S FORMULA. 



In Fig. 146, A D B represents the axis of a strut, carrying a load 

 P, originally applied in an axial direction, and causing the pillar to be 

 deflected at its centre D an amount d. 



Let y be the deflection at any point 0. At 



M= Py (see " Deflection of Beams ") 

 _EI 



therefore, Py _ _ d*y 



EI~ dx*' 



Now, integrate, Py dy _ _ dy d 2 ;/ 



El dx ax dx* 

 P y* 

 Wl~2 

 dy P S* 



tx = - c = m ~2 y = s - 



Therefore, J 



Integrating again, x = ^ET Sin -\ y_ + A> 



* P d 



when x = o, y = o, and 



therefore, A = o 



So that /~ J y 



this is the equation to the sinusoid, or elastic curve. 



sothat 



that is to say L\/~P~= 



~*E~I 2' 



P ]P__ v* 

 4 E I ~ 4" 



or 



which is Euler's equation. 



