CHAP. XIV.] THE BRACED ARCH. 257 



dimensions of the centre line, we may, if we choose, suppose 

 the depth of the arch to vary above and below the centre line 

 equally, from the crown to ends. The lever arms of the pieces, 

 and hence their strains, will be different, bnt V and H are the 

 same as before. Thus, whatever the shape of the arch, we can 

 easily find the strains both by diagram and calculation. If we 

 draw a line through A and the hinge at crown, we may easily 

 prove that the greatest vertical ordinate between this line and 

 the arch is 



where r is the radius. 



Now if the depth d of the arch is made greater than this or- 

 dinate, it may be shown that both flanges will always be in 

 compression. This condition serves, then, to determine the 

 proper depth of circular arch, which should not be less than 



+ - r. 



It is unnecessary to give here an example.* The method is 

 so simple that the reader will find no difficulty in applying the 

 principles above to any case. He will do well to calculate or 

 diagram the strains in an arch similar to that shown in the Fig. 

 for comparison with the two cases which follow. 



159. Arch hinged at Abutments continuous at Crown. 

 If we suppose the hinge at the crown removed those at the 

 abutments being, however, retained then, for any position of 

 the weight, the resultant at each end must for equilibrium 

 pass, as before, through the end hinges. In the preceding 

 case, a, for load on left half, was always to be found atintersec 

 tion of weight with the line through B and hinge at crown, and 

 was therefore fully determined. Now, however, #, the com- 

 mon intersection of weight and resultant abutment pressures, 

 has a different position, and hence the resultants and horizon- 

 tal and vertical reactions are different. 



If we can find or know the locus or curve in which this point 

 a must always lie, we can easily find, as before, the resultants 

 or reactions by simply prolonging the line of direction of the 

 weight till it meets this locus, and then drawing from the point 



* See Note to this Chap, in Appendix. 



