DEFLECTION OF CANTILEVER AND SIMPLE BEAMS 



Fob 



67 



and the moment at the load is 



Also, the area of the moment 



diagram between the load and one end is - , and the distance 



2 

 of the centroid of this segment from the end is x = - a. Hence the 



o 



deflections of the ends from the tangent at the point of application 

 C of the load are p/i8 ^ 



and 



Pab* 



ZEIl 3 Ell 



and the deflection of C below the level of the supports is 



b 2 pab 



(56) 



d = 



3 Ell 3Ee 



where p denotes the maximum fiber stress. 



41. Simple beam bearing uniform load. For a simple beam uni- 

 formly loaded the moment diagram is a parabola, the maximum 



wl 2 

 ordinate being h -Z-- 



o 



From article 17, the 

 area of this parabola is 



2 wl* 7 _^_ 8 

 "5" 8 ' = 12' 

 To apply the general 

 formula for deflec- 

 tion, consider d as 

 measured from one 

 end A to the tan- 

 gent at the center 

 C (Fig. 69). Then, 

 since the area of one 



FIG. 



wl 3 



half the moment diagram is , and the distance of the centroid of 

 this half from a vertical through A is X Q = - - = ^ the deflection is 



(57) 



= - 



To express the deflection in terms of the maximum fiber stress p, 

 make use of the relation ^ = M = ~ Then, replacing -, in the 



