DESIGN OF FIXED ENDED ARCHES BY THE ELASTIC THEOEY 137 



total sum of deflections, both horizontal and vertical, for the entire 

 length of any fibre of the rib, must be equal to zero. By the com- 

 mon theory of flexure it can be shown that: — 



D, = 4,..P-«=4,^^, (7) 



D,=2^,..P'y=4,^. (8) 



Here D^. and D^ represent the total vertical and horizontal de- 

 flections respectively for any given segment or part of a beam or rib 

 depending upon what limits of summation are taken, n, P', M, E 

 and I have the same significance as above while x and y are the rec- 

 tangular coordinates of any section. In plate I the origin of coord- 

 inates is taken at Y' and the axis of X coincides with VY ', the chord 

 of the neutral curve of the rib. 



If the total summation of deflections for the entire length of rib 

 be taken, we have: — 



D,=2^,;.P>=^2^,M,r=0, (9) 



EI 



n 

 EI 



D, = 2^,/.P'y=^2^,My=0. (10) 



Hence:— Sy,Ma;=0, (11) 



2^,My/=0. (12) 



From equations (11) and (12) it follows that if the total sums 

 be taken of the products obtained by multiplying the bending 

 moments at different sections respectively by the coordinates of such 

 sections that such sums must be equal to zero. 



When the ribs are assumed continuous with fixed ends the con- 

 ditions therefore upon which the design depends are: — 



