60 MOVING LOADS ON BEAMS 



Differentiating (16) and placing derivative of M with respect to x 

 equal to zero, we have after solving 



x=o (17) 



Therefore the maximum moment at any point in a beam will occur 

 when the beam is fully loaded. 



The bending moment diagram for a beam loaded with a uniform 

 moving load is constructed as in Fig. 38. 



To find the position of the moving load for maximum shear at any 

 point in a beam loaded with a moving uniform load, proceed as fol- 

 lows : The left reaction when the end of the moving load is at a dis- 

 tance x from the left reaction, will be 



*, = i P (15) 



and the shear at a point at a distance a from the left reaction will be- 



which is the equation of a parabola. 



By inspection it can be seen that $ will be a maximum when 

 a = x. The maximum shear at any point in a beam will therefore 

 occur at the end of the uniform moving load, the beam being fully 

 loaded to the right of the point as in (a) Fig. 39 for maximum positive 

 shear, and fully loaded to the left of the point as in (b) Fig. 39 for 

 maximum negative shear. 



If the beam is assumed to be a cantilever beam fixed at A, and 

 loaded with a stationary uniform load equal to p Ibs. per lineal foot, and 

 an equilibrium polygon be drawn with a force polygon having a pole 

 distance equal to length of span, L, the parabola drawn through the 

 points in the equilibrium polygon will be the maximum positive shear 

 diagram, (a) Fig. 39. The ordinate at any point to this shear diagram 

 will represent the maximum positive shear at the point to the same 

 s^ale as the loads ( for the application of this principal to bridge trusses 

 see Fig. 50, Chapter X). 



