CHAP. Xm.] ANALYTICAL FORMULA 241 



156. Thus we see that, as in Art. 150, a few short and sim- 

 ple formulae, which may be written on a piece of paper the size 

 of one's hand, are all that we need for the complete solution of 

 any case of level supports whether the spans be all equal or 

 the end ones only different, or all different ; whether the girder 

 merely rest on the end supports or be fastened horizontally at 

 one or both ends. We have only to remember that a positive 

 moment causes tension in upper flange at support, and there- 

 fore compression in lower; inversely for negative moment. 

 Also, that a positive shear acts upwards, and a negative shear 

 downwards. Also, that both moment and shear are positive at 

 supports of loaded span, and alternate in sign both ways. This 

 is all that we need to form properly the equation of moments 

 at any apex, and determine the quality of the strains in flanges 

 and diagonals. We can thus solve any practical case of framed 

 continuous girder which can ever occur with little more diffi- 

 culty than in the case of a simple girder. 



Thus, for the span D E (Fig. 87) we have only to find the 

 moments at D and E due to every position of P in the span 

 D E, and the corresponding shears at D. These once known, 

 and, as we have seen, they can be easily obtained from our 

 formulae, we can find and tabulate the strains in every piece 

 due to each weight, as shown in Art. 127. An addition of these 

 strains gives, then, the maxima of each kind due to interior 

 loading. 



We have, then, to find, in like manner, the strains due to the 

 two cases of exterior loading as represented in Fig. 87, or else due 

 to each exterior span loaded, making a column for each span. 

 From the columns thus obtained, we can deduce the dead load 

 strains, and then finally the total maximum strains of each kind 

 for every piece. [See, for illustration of the above, Art. 127.] 



Thus, the whole subject is solved with the aid of but four 

 simple formulae, and for a problem generally considered impos- 

 sible by reason of its " complexity," our results will, we trust, 

 be found sufficiently simple and practical. 



In view of the fact that the necessary formulae for practical 

 computations have been often given in the later works of 

 French and German authors, although perhaps never before 

 in so compact and available a shape as above, it is indeed sur- 

 prising that they should have been so completely ignored bj 

 English and American writers. 



The tables and formulae which we have given will, we trust, 

 16 



