FLEXURE OF BEAMS 



87 



and if the values of Jf a corresponding to each value of x from d to 

 / are laid off as ordinates, we obtain the straight line A'B', which 

 therefore represents the variation in the bending moment at the 

 point A as the unit load moves from B to A. Similarly, if the unit 



load is on the left of A, J/ (l = 



x (I- d) 

 I 



which is the equation of the 



straight line O'A'. At D' both lines have the same ordinate, namely, 

 J ' K = - The influence line for bending moment is therefore 



tla- l.i cken line O'A'B'. 



From this construction, it is obvious that the ordinate to the influ- 

 ence line at any point D represents the bending moment at A due to 

 a unit load at D. Thus, as a 



unit load conies on the K'um flf A _ [ B 



from the right, the bending ]~ 

 moment at A increases from 

 the value zero for the load 

 at B to the value A'E for the 

 load at A, and then decreases 

 ; i-_ r a in to the value zero at 

 O. Therefore, having con- 



structed for a unit load the influence line corresponding to any given 

 point A, the moment at A due to a load P is found by multiplying 

 P by the ordinate to the influence line directly under P. 



Problem 95. Find the i><irion of a system of moving loads on a beam so that 

 th- I't-ndiim moment at any point A shall be a maximum. 



Solution. Let O'A' If be the influence line for bending moment for the point A, 

 ami let the loads on each side of A be replaced by their resultants PI and P 2 

 08). Tin n. if y , siiid y 2 are the ordinates to the influence line directly under 

 PI and P 2 , the moment at A is 



FIG. 68 



-V,, = 



+ P 2 y 2 . 



Now, if the loads move a small distance dx to the left, the moment at A becomes 



M a + dM a = P! (yi - dx tan a) + P 2 (y 2 + dx tan/3). 

 Therefore, by subtraction, 



dM a = - PI dx tan a + P 2 dx tan /9, 

 and hence 



- = - PI tan a + P 2 tan. 

 dx 



