138 UNIVERSITY OF COLOEADO STUDIES 



2:m=o, \ 



2My=0, (13) 



n \ 



— -=:a constant. I 

 EI / 



Every polygon similar to ?/iBIJKm' or KjEFGHIv^ is an equi- 

 librium polygon for some loading. Their vertical interscepts in 

 every case are proportional to moments. Of all these polygons, one, 

 namely KjEFGHKj has the same closing line and the same hori- 

 zontal thrust as the rib. This polygon is known as the true equilib- 

 rium frame. 



It can be shown from the principles of equilibrium frames 

 (graphic statics) that the bending moments at all points in any 

 arched rib are proportional to the vertical interscepts at such points 

 between the rib neutral curve and the true equilibrium frame. This 

 principle holds both for free and fixed ended ribs. Let M be the bend- 

 ing moment at any point in an arched rib. Let Mj be the moment 

 intercepts for the true frame KjEFGHKj and M,i the interscepts for 

 the rib. Then from what has just been stated : — 



(14) 



(15) 

 (16) 



(17) 



2^,My=2^,M,2,-S^,M„2/=0. (18) 



The last relation, equation (18), is used in developing the de- 

 flection polygons for the arch rib and the trial frame in order to de- 

 termine the true pole distance and true interscepts of the true frame. 



In designing fixed ended arches by the elastic theory the proced- 

 ure in applying the above analysis is as follows:— a true equilibrium 

 frame must be found and also a closing line common to it and the 



