282 SUPPLEMENT TO CHAP. XIV. [CHAP. H 



O = H cos V sin 0, O' = H cos + V sin ) f ^^ 

 N = H sin + V cos 0, N' = H sin V" cos ) 



M = H(ft-y)-V(a-*) t M' = H (A - y) - V (a + *) . . (19) 



In the case of a circular arc, a = r sin a, h = r (1 cos a), x = r sin 0, 

 y = r (1 cos 0), and hence 



M = H r (cos cos a) V r (sin a sin 0) ) ,., 



M' = Hr (cos cos a) V'r (sin a + sin 0) ) 



4. Intersection Line. We call the locus of d [PL 23, Fig. 91], or 

 the curve edeiJc, the intersection line. 



If now there are three hinges, one at crown and one at each abutment, 

 then the resultant for each half must pass through the crown O. If, there- 



fore, for the crown (a;=0, y=0), we make in (19) M'=0, we have H=V'~, 



m 



or inserting the value of V from (16), 



_ P (a T g) , 91 v 



~ ......... 21 



If the load lies to the left of O, then only the resultant R' acts upon the 

 right half, and must pass, as above, through the crown O. The point d 

 lies then always upon B O or A O prolonged. Hence, the intersection lines 

 are two straight lines, which pass through the crown and ends. 



5. Parabolic Arc concentrated Load. For a parabola we 



have y = .x t \ hence, d y = - x d ar, and, approximately, d s =d x. 

 a . a 



(a) Change of direction of tangents. 



Inserting this value of y in equations (19) for M and M', we have from 

 equation (13), Art. 1, since rd(j) = ds = dx for the change of direction of 

 the tangent at any point before and after flexure, 



Integrating this, we have for the three segments A E, E O and O B, 



^~ X \ ~fTa s )~ ' \ ~2a; 



/I 1 Vaxll + | + A' . . (22) 

 \ 3 2 / \ 2a) 



f s?-l + A' 



where A, A', A' are constants of integration, to be determined by the as- 

 signment of the proper limits. Thus, if we make x = z, the two first of 

 equations (22) are equal; hence, 



-Vazll-^- 

 \ 2a 



