88 



STRENGTH OF MATERIALS 



For a maximum value of M a , = 0, in which case 



ax 



This equation may be written 



P 2 tan/3 = PI tan a. 

 . A'C' , A'CT 



or 



p l 



from which, by composition, 



which is the criterion for maximum moment at A. Expressed in wor<K ' 



at any point A is a maximum when the unit load on the whole span i* equal to the 



unit load on the smaller segment. 



75. Influence line for shear. To obtain the intluen.v lin, 

 shear, let I, d, and x have the same meaning as in the pieeedintf 



article. The shear at any 

 point A is equal to the re- 

 action at 0, and for a unit 

 load this reaetion is 



-d- -,/" ~~cf 



l-x 



I 



If, then, the values of /, 



all values of x from d to I 



are laid off as ordinates, the locus of their ends will be the straight 

 line B'A' (Fig. 69). Similarly, for a unit load on the left of .4 the shear 



FIG. 69 





at A is negative, and its amount is /?,= - which is the equa- 



tion of the straight line O'A". Since the slopes of the two lines 

 A'B' and O'A" are equal, these lines are parallel. The influence line 

 for shear is, then, the broken line 0'A"A'I>'. 



As a load comes on the beam from the right the shear at A gradu- 

 ally increases from the value zero for the load at B to the value . I ' I-' 

 for the load just to the right of A. As the load passes A the shear at 

 this point suddenly decreases by the amount of the load, thus becom- 

 ing negative, and then increases until the load reaches O, when it 

 again becomes zero. Consequently, the shear at A, due to a load /' at 



