CHAP. IV.] 



SUPPLEMENT TO CHAP. XIV. 



301 



19. Intersection Curve. From Art 13 we have 

 _ V r sin j8 M 

 H 



Hence, by substitution of the values of V, H and M , 



^-iol 1 - 



120 (a 2 - 2 a ft - j8 2 ) 



a 4 (a + 



y = 



-30' 



1 



(47) 



(a + ) A A 2 J 



which is the equation given in the text, page 265, for which a table is there 

 given. 



SO. Direction Segments. It will in the present case be found most 

 convenient to determine the directions of the resultants by Ci and C 2 equa- 

 tion (34). 



Thud, Gl = - 5^, c 2 = - ^. 



But Mj = Mo H h + (P V) a P s, M 2 = M H h + V a. 

 We have, by aeries, then the approximate formulas, for small central angles, 

 not over 40, 



2 h r 45 1 



\a-5z 



A A 2 J 15 (a + z) L A 



where positive values of d and e 2 are laid off upward above, negative values 

 downward below, the centres of gravity of the end cross-sections. 



From the preceding tables we can calculate easily in any case H and V and 

 MO, and thus check the results obtained by the method of Chap. XIV. The 

 formulas above for Ci and c 2 do not admit of tables, nor, in fact, are such 

 needed. They are sufficiently simple for ready insertion. 



Thus, by the aid of our tables, having computed V and H, and, if necessary, 

 MO and , we can by the method of moments, as explained in Chap. XIV., 

 Art. 162. readily calculate the strains in the braced arch, whether continuous 

 at crown and fixed or hinged at the ends, or hinged at both ends and crown. 



Direction Curve. We may also plot the direction curves, from the 

 following Table calculated by Winkler from the exact formulas, and which 

 applies to central angles up to 90 : 



