DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEORY 143 



then drawn in the following manner: From B draw BI parallel to 

 the ray CB to terminate in a point I upon the line of action I — 18 

 of the load acting at the centroid of the part 18 of the rib. From I 

 draw the side IJ parallel to the ray C — 17 to meet the line J — 17 in 

 the point E, etc. Finally connect the points B and K. 



Lines like Ic in the polygon BIJKB or t2 in the rib are called 

 interscepts while <iK and 2V' are known as the lever arms re- 

 spectively of the interscepts. Ic and t2 are proportional to bending 

 moments. Hence products like IcXcK represent quantities like Ma*. 

 Similarly Ic multiplied by a vertical ordinate would represent a 

 quantity like My. 



We must now refer to Table II. Column 1 indicates the point 

 or part of the rib to which the figures to the right refer. Column 2 

 gives the arch interscepts in feet. They are the vertical interscepts 

 like t2 at the respective points of the rib between the chord W and 

 the neutral curve. They are scaled from Plate I. 



Since 2M=0 for fixed ended arches the closing line KjKg of 

 the rib can now be drawn. Kiy=K2V' = the average of the sum 

 of the arch interscepts = 21.782 feet. 



. . n 



This closing line KjK, satisfies for the rib the conditions -^^ = 



constant, SM=0, 2M,/'=0, and 2My=0; because the rib neutral 

 curve is symmetrical both with respect to the vertical and horizontal 

 and because KjK., is horizontal. It is the line of action for the hori- 

 zontal thrust in the rib. It is also the closing line of the true or 

 equivalent equilibrium frame EFGH. 



The trial polygon BIJKB is now operated upon to find the true 

 pole D of the true equilibrium frame KjK^HGFE. Columns 3 and 

 4 of Table II give respectively the interscepts and lever arms of the 

 points of the trial polygon. These distances are scaled from Plate I. 



If this trial polygon were the true polygon, a closing line could 

 be found which would make the following relations hold : — 



2M=0. 



2M«=:0. 



2M7y=0. 



