DEFLECTION OF CANTILEVER AND SIMPLE BEAMS 63 



37. Cantilever bearing concentrated load. For a cantilever bear- 

 ing a concentrated load at the end, the moment diagram is a 



triangle, as shown in 

 Fig. 63. The area 

 of the moment dia- 



gram 



is therefore 



(38) 



FIG. 63 



d - 



~ 



PI 2 



A = j and the dis- 

 tance of its centroid 

 from the free end 



2 



is X Q = - I. Therefore 

 o 



the deflection at the 

 free end is 



El 3 El 



The deflection d may also be expressed in terms of the stress 



on the extreme fiber. Thus, since ^- = M = Pi, by substituting 



this value of PI in 



#m 



the expression for c?, it 

 becomes 



(39) 



d = 



3Ee 



Also, the angular 

 deflection at the load 

 is found to be 



(40) tan </> = 



El 



Fl* 



2 El 



FIG. 64 



If the load is at a distance a from the fixed end and b from the 

 free end (Fig. 64), then the deflection at the load, as shown above, is 



Pa* 

 (41) d-- 



