DEFLECTION OF CANTILEVER AND SIMPLE BEAMS 65 



7* 72 



From the relation = M = the expression for the deflection 

 in terms of the maximum fiber stress p is found to be 



(48) -, ** 



Also, the total angular deflection at the end of the beam is 

 (49) 



, A wl 3 



tan = = 



Y El 



6 El 



39. Cantilever under constant moment. If a cantilever is sub- 

 jected to a couple, that is, a pair of equal and opposite parallel 

 forces, as shown in Fig. 66, the moment is constant for the entire 

 length of the beam. The moment diagram is therefore a rectangle 

 of area Ml, and the de- 

 flection at the free end is 



1 



(50) d = Aoc Q 



Ml* 

 = 2 El' 



The angular deflection 

 at the free end in this 

 case is 



(51) ten* = ^L 



Ml 



40. Simple beam bearing concentrated load. To apply the deflec- 

 tion formula to a simple beam, the deflection must be measured at 

 one end A from a tangent at the middle C. For a concentrated 

 load P at the middle (Fig. 67), the area of the moment diagram 



P/ 2 2 I I 



from A to C is A = =--, and x. = - ^ = 5- Hence, in this case 

 lb o A o 



(52) 



Pf 



