CHAP. XTTI.] ANALYTICAL FORMULAE. 225 



where %' = P k for concentrated load, and g[ = ~ for uni- 



2i \ 



form load. For any other span at the right support 



g , _ M m - M m _ t 



lm-i 



Thus, S m and S' m are the shears at any support just to right and 

 left of that support respectively. The reaction at any support 

 is then 1^ = S.a + S' m . 



The moments, then, at two successive supports being known, 

 we can readily find the shear at any support, and these two, 

 moment and shear, we repeat, are the quantities required for 

 calculation. The reactions, and the tables for the reactions 

 above, are only useful as enabling us to find the shear. It is 

 this last, together with the moment at the support, which gives 

 us the moment m at any point of the span in question, as is 

 evident from the above equations. It is only in the case of the 

 simple girder that the reactions at the ends are the same as the 

 shears. In the continuous girder only the latter should be 

 used, except for ends of end spans, where the two are identical. 

 "We have only to remember, then, that the shear at any support 

 is the algebraic sum of all the reactions and loads from that 

 support to the nearest extreme end, and then, knowing these 

 reactions and loads, the determination of the shear is easy. 



We might give tables for shears directly, as above, for reac- 

 tions; but this is unnecessary. Having taken the reactions 

 from our tables already given, and found the moments either by 

 our formulae or tables, we can then find the shears both by means 

 of the reactions and also directly from the moments them- 

 selves, and thus check at once the accuracy of our determina- 

 tion of both. From what has already been given, the reader 

 can easily construct tables of shears similar to those already 

 given for reactions for himself, if desired. 



149. Recapitulation of Formulae Continuous Girder 

 over any Number of Level Support*, all Spans equal.* 

 for notation, see Art. 130, Fig. 89. 



* These formulae are given in similar form in Winkler's "Der Lehre von 

 der Elasticitaet und Festigkeit," Art. 144, p. 122. They were also indepen- 

 dently deduced by Mr. Merriman, to whose kindness we are indebted for much 

 of this chapter. The above methods of tabulation were communicated by 

 Mm, and are given in no treatise upon the subject. 

 15 



