CHAP. V.] MOMENT OF EOTATION PARALLEL FORCES. 49 



ment of L 7 is given by the ordinate o y to the corresponding 

 polygon, arid we may consider L 7 as acting at the point of 

 intersection a of the side 7 8 with A' B' (Art. 38). In the same 

 way Li' 3' acts at b. We may unite both these reactions and 

 find the point of application of their resultant c, by laying off 

 in force polygon (b) 7 b equal to L' 3', and then constructing 

 the corresponding equilibrium polygon e a d o. The resultant 

 R passes through c. This construction remains the same evi- 

 dently, even when the points a and b fall at different ends of 

 the beam, as may indeed happen. The components will then 

 have opposite directions, and must be subtracted in order to 

 obtain the resultant. 



The total moment of rotation at y is proportional to the sum 

 of m n and o y. The greatest strain is where this sum is a^ 

 maximum. In order to perform this summation and ascertain 

 this point of maximum moment it is advantageous to construct 

 another polygon instead of A' 1" 2", etc., whose closing line 

 shall coincide with A' B'. This is easy to do, by drawing in 

 force polygon (a), I/ C' parallel to A' B', and taking a new pole 

 C' the same distance out as before, that is, keeping H constant, 

 and then constructing the corresponding polygon A' 1' 2' 3', etc. 



Thus the ordinate p y gives the total moment at y. We can 

 make use here also of the principle that the corresponding sides 

 of the two polygons must intersect upon the vertical through 

 A' (Art. 26), We have thus the total moment at any point, and 

 can easily determine the point of maximum moment or cross- 

 section of rupture. This point must necessarily lie between 

 the points of maximum moments for the two cases, or coincide 

 with one of them. In the Fig. this point coincides with the 

 point of application of P' 2 . 



45. Case of Uniform Load. If the continuous load is uni- 

 formly distributed we can obtain the above result without 

 being obliged to draw the curve. As in this case we have a 

 very short construction for the determination of the point of 

 greatest moment, it may be well here briefly to notice it. 



If we erect ordinates along the length of the beam as an axis 

 of abscissas, equal to the sum of the forces acting beyond any 

 cross-section, the line joining the end points of these ordinates 

 has a greater 01 less, inclination to the axis according as the 

 uniform load is greater or smaller. At the points of applica- 

 tion of the concentrated loads this line is evidently sfdfted 



