ARCHES AND ARCHED RIBS 187 



ratio, say r. The problem then is to find this ratio r in which the 

 ordinates to the equilibrium polygon must be reduced to give the 

 linear arch. 



The condition that the span is unchanged in length, derived in 

 the preceding article, is 



iii which z represents the ordinate CE to the rib, and ds an element 

 of the rib. Since the bending moment M is proportional to the verti- 

 cal intercept between the linear arch and the center line of the rib, 

 this condition may be written 



/ 



==-'*-; 



or, since E may be assumed to be constant and 

 BC = BE - CE = BE - z, 



this condition becomes 



J(*p) & = 0> 



which may be written 



If r denotes the ratio in which the ordinates to the equilibrium 

 polygon must be decreased in order to give the linear arch, then 



BE 



r = 



DE 



and consequently the condition becomes 



whence 



CPE- 

 J I 



DE-z, 



as 



This expression for r can be evaluated graphically by replacing 

 the integrals by summations and calculating the given functions for a 



