THE BRACED ARCH. [CHAP. XIV. 



2 h T 45 d*a~\ 



I a -\- o x . I 



15 (a x) I 4A a 



2A 

 15 (a + 



[ 



[A negative result indicates that the distance is to be laid off 

 below the centre of cross-section.] These formulae are easy of 

 application, and sufficiently exact for arches whose rise is small 



compared to the span ; when - is, say, not greater than -j^. 



- a 



All the above f ormulse are for constant cross-sections. Exact 

 formulas for variable cross-section give results but little less, 

 and are much more complicated. The effect of using the above 

 formulae is therefore, merely, to increase slightly the coefficient 

 of safety. In other cases we can easily plot the direction curve 

 from the Table of Art. 20, page 301. It is well to do this in 

 all cases, and thus check our values for c, and c v 



161. We are now able to determine readily and accurately 

 the strains in the various pieces of braced arches hinged at 

 crown and abutments, and hinged at abutments only. We 

 have only to construct in each case the reactions at the abut- 

 ments, as explained in Arts. 158 and 159, Figs. 90 and 91, and 

 then, by the method already detailed in Arts. 8-13, we can fol- 

 low these reactions through the structure, and thus iind the 

 strains in each piece due to every position of the load. We 

 may also, having found the reactions for given position of 

 weight, calculate the strain in each piece by moments. 



For the case of the arch continuous at the crown and fixed at 

 the abutments^ we must remember that we have also a moment 

 at each end tending to cause either tension or compression in 

 the inner flanges according as it is negative or positive. The 

 case is precisely analogous to the continuous girder, or girder 

 fixed at ends. As in that case [see Fig. 77, Art. Ill] the 

 moment at one end, as B, was the product of H into the vertical 

 distance B D, so here the moment at A (Figs. 92 and 93) is the 

 product of H into c 1} found by the formulae above. This 

 moment can, then, be easily found when c^ and H are known. 

 We can then lay it off, according to the directions of Art 125, 

 for "passing from one span to another of a continuous girder," 

 and thus commence our diagram of strains; or we can cal- 

 culate the strains by the method of moments. 



