156 SHEARS AND MOMENTS. Art. 03. 



because the external forces form a couple whose moment is P/, 

 for any section between the loads (30) . 



Fig. 101. This is perhaps the commonest case. The reac- 

 tions are apparent. The shear at the middle 2^ %wL = % 

 wL y 2 wL = Q, and must vary uniformly between the ends and 

 this point. The moment is, of course, a maximum at the middle 

 and is 



RiXV2L-y 2 wLxyL = 7tf max = y s wL* = y 8 WL (42) 



This value is the same as the general equation will give 

 when x = %L. 



Equation (42) is important because it is^used very fre- 

 quently. y s WL is the middle ordinate of the parabola. 



If a girder 25 ft. long carries a total uniform load of 

 4000 Ibs. per lineal foot, 



w =:4000 Ibs. W = 4000X25 100000 Ibs. L = 25 ft. 



H 1 = R 2 = max = 50000 Ibs. 



Shear at middle = 50000 4000X12% = 0. 



^max = y s WL = %X 100000X25 = 312500 ft. Ibs. 



Moment at 5 ft. from one end = 50000X5 5X4000X2%=-= 

 200000 ft. Ibs. 



Comparing the moment produced by a uniform load with 

 that produced by a concentrated load at the middle, it is found 

 that a beam will carry twice as much load when it is uniformly 

 distributed as when it is concentrated half way between the 

 supports. 



Fig. 106. Under the uniform load, the moment diagram is 

 parabolic. If the load were concentrated at the middle of CD, 

 the diagram would be FGH; if at C and D, it would be FKNH. 



94. Combinations of Concentrated and Uniform Loads on 

 Cantilever Beams and Beams Supported at the Ends. Since 

 the weight of the beam itself is usually taken as a uniform load, 

 the case of a beam carrying concentrated loads only does not 

 occur. 



Finding the shears and moments in a cantilever beam for 

 any combination of uniform and concentrated loads, applied on 

 any part of the beam, is simply a matter of adding together tli 

 n -suits gotten for each load considered separately, or applying 

 the equations of equilibrium. This is also the case for combina- 

 tions of loads on beams supported at both ends, but, in general, 



