FLEXURE OF BEAMS 



89 



any point C, is found by multiplying P by the ordinate to the influ- 

 ence line at C', directly under C. 



Problem 96. Find the position of a system of moving loads on a beam so that 

 the shear at any point A shall be a maximum. 



Solution. Let the influence line for the point A be as represented in Fig. 70. 

 Also let P! and P a be two consecutive loads, d the distance between them, and P / 

 the resultant of all the loads on 



P, 

 A 



the beam. Since A'E is the 



maximum ordinate to the influ- 



ence line, the maximum shear 



at A must occur when one of 



the loads is just to the right of 



A. Suppose the load PI is just 



to the right of A. Then as P! 



passes A the shear at A is sud- 



denly decreased by the amount 



PI. If the loads continue to 



move to the left until P 2 



reaches A, the shear is gradu- 



ally increased by the amount 



P'd tana, since the ordinate 



under each load is increased by the amount d tan a. Consequently, either 



PI at A will give the maximum shear at this point according as 



B 



K 



B' 



FIG. 70 



or 



or, since tana = -, according as 



P'd tan a ; 



By means of this criterion, it can be determined in any given case which of two 

 consecutive loads will give the greater shear at any point. 



76. Maxwell's theorem. When a load is brought on a beam it 

 causes every point of the beam to deflect, the amount of this deflec- 

 tion for any point being the corresponding ordinate to the elastic 

 curve. If, then, a number of loads rest on a beam, the deflection at 

 any point of the beam is the sum of the deflections at this point due 

 to each of the loads taken separately. 



For example, if two loads P l and P 2 rest on a beam at the points 

 J and B respectively, the deflection at one of these points, say A, is 

 composed of two parts, namely, the deflection at A due to P l and the 

 deflection at A due to P v Similarly, the total deflection at B is com- 

 posed of the partial deflections due to P l and P 2 respectively. 



