CHAP. Xm.] ANALYTICAL FOKMT7L2E. 207 



For supports 3, 4 and 5, m = 3, and 



or since in this case the moments equally distant each way 

 from the middle are equal, this last equation becomes 



M, + 4 M 4 + M 4 = ^1\ 



We have therefore three equations between three unknown 

 moments, M^, M 3 and M 4 , and by elimination and substitution 

 can easily find 



e 



If, as in our example of Art. 127 in the preceding chapter, 

 we take u = 1 ton per ft., I 80 ft., then u 1? = 6400, and the 

 moment at the fourth support becomes 540.8. If the height of 

 truss is ten feet, this gives [Fig. 88] 54.1 tons strain in the 

 upper flange A a. By reference to our tabulation, Art. 127, 

 we see that this agrees closely with strain in A 1 due to uni- 

 form load, found in a manner entirely different, viz., by sum- 

 mation of the strains due to first case of loading, and the several 

 loads in the span itself, and serves therefore as a check upon 

 our results. 



133. Triangle of Moments. For the benefit of the practi- 

 cal engineer, who may object to the algebraic work involved in 

 elimination of the unknown moments from the equations above, 

 when the number of spans is great, we offer the following tabu 

 lation, from which he may easily and directly determine th 

 moments at the supports for any desired number of spans 

 without formulcs or calculation. 



Thus, if we were in the above manner to find the moments 

 for a number of spans, and tabulate our results as given in the 

 annexed table, an inspection of the table will show us that we 

 can produce it to any extent desired without further calcula- 

 tion. 



