36 DEFLECTION OF BEAMS. 



Therefore we may write 



= - 



El da? da? 



Take the simplest and most frequently occurring case in 

 testing, where the beam is supported at the two ends and 

 loaded in the centre with a load W. The reactions at the 



W 



supports are - , and the bending moment of any section 

 2 



distant x from is 



AT 

 M = 



Integrating 



V rd?y W 



"ap-r 



we have 



V1 dv W , Wo 2 , r 

 JL 1 = > t <c - -- h Vi 

 </# 4 4 



C, being a constant of integration. When x = 0, the 

 tangent is horizontal, so that -^ = 0, C, = 0. . 



(JL QC> 



Therefore E I ^L= ^ (ix 



^~ 



Integrating again 



dx 4t 

 Wlx* War 1 



c 2 being again a constant of integration. When 



x = o, y = o, and c 2 = o, 

 therefore we have 





W (la? x 3 

 ~"ET\ 8 12 



If we put x = --, y becomes the deflection at the centre, or 



W / P P_\ 

 "El ^82 "" 96) 



y 



OJJ X 



W/ 3 



< XXVI -> 



That is, for a horizontal beam of span I, and moment of 

 inertia I, formed of a material whose modulus of elasticity 

 is E, and loaded centrally with a weight W, the deflection 

 at the centre, 8, is given by the above formula. 



