CHAP. Xm.] ANALYTICAL FORMULA. 223 



above are so simple, and with a little practice so readily worked, 

 that tables are scarcely needed. 



147. Reactions at Supports for concentrated Load in 

 single Span. For the reactions we have the following formulae: 



1st. Abutment reactions. 



When the end span contains the load, that is, when r = 1 or 

 *=?* 



When the load is not in the end spans, i.e., when r > 1 and 



r <s, 



R M * R M " 



> *W - 



2d. Reactions at supports of the loaded span itself (not end 

 span), 



= 6 + P (1 - 3 & + 3 & - P), 



3d. For all other reactions. 



I ' 



The above formulas, in view of what has been said in Art. 

 145, are sufficiently simple to need no illustration. 



For load in fourth span of seven spans we find easily for the 

 reaction at left support, 



R 4 = - L_ [2911 -3k- 6387 P + 3479 

 .2911 L 



This can be put in working order as explained in Art. 145, 

 and the reader can check the results which we have given in 

 the example of Art. 127 for himself. 



A triangle and two subsidiary tables for the reactions at the 

 supports of loaded spans may be formed similarly to the trL 

 angle and tables of Arts. 142 and 143. We leave this for the 

 reader to accomplish for himself, if thought desirable. 



1 18. Shear at Supports of loaded Span. We are now in 

 possession of all the formulae necessary for the complete solution 



