CHAP. V.] MOMENT OF KOTATION PARALLEL FOKCES. 51 



gon, and unite the forces and moments thus found with those 

 already existing. 



In PI. 7, Fig. 32, we have assumed a new force P\ near the 

 left support. The force polygon is 0' 1' C', the pole distance 

 being taken the same as before. For any one position of this 

 force we have then the equilibrium polygon A' 1' B", and 

 drawing a parallel C' L' to A' B" we obtain the reactions 0' L' 

 and L' 1', which must be added to the reactions already ob- 

 tained. 



If now we take a section y between P\ and the point of max- 

 imum moment 2 before found, the sum of the forces either 

 side of this section undergoes the following changes : Upon 

 the side where P^ lies, and the point 2 does not lie, where 

 therefore the sum was originally an upward force, we have the 

 downward force L' 1' (equal to algebraic sum I/ 0' + 0' 1'). 

 The sum of the forces at the section, or the shearing force, is 

 therefore diminished. 



The total rotation moment is, however, increased by the 

 amount indicated by ra n. Both changes, that of the sum of 

 the forces and the moment of rotation, increase as P\ ap- 

 proaches y, and are therefore greatest when P\ reaches y. 



If P\ passes y, this point is in the same condition as z with 

 reference to the former position of P\ ; that is, the force and 

 point 2 are now both on the same side of the section. For z, 

 then, the original downward force to the left is increased by the 

 force L' 1'. To the right the upward force is increased by 1' I/. 

 In like manner the moment of the forces beyond z is increased 

 by the amount indicated by op. This change is greatest when 

 P' t reaches z. 



Therefore when a load passes over the beam the sum of the 

 shearing forces is diminished in all sections between it and the 

 original point of greatest moment, and increased in sections be- 

 yond this point, while the moment of rotation, or bending 

 moment, for all cross-sections is increased. These changes 

 moreover increase for any section as the load approaches that 

 section. The shear at any point is therefore least, and the mo- 

 ment greatest, when the load reaches that point. As soon, how- 

 ever, as the load passes this point, the shear passes suddenly 

 from its smallest to its greatest opposite value, and then dimin 

 ishes as the load recedes, together with the njoment of rotation. 

 On the other side of the point 2 of original greatest moment, 



