392 NOTE TO CHAP. XTV. [APPENDIX 



A comparison of this table with the preceding shows that we 

 have gained nothing by introducing two end flanges at A at 

 each end, and pivoting the arch at the extremities of the centre 

 line. We have indeed slightly diminished the strains in the 

 lower flanges E and F, as also in the bracing, but the other 

 strains are much greater than before a result which might 

 have been anticipated, since the effect of hinging at the centre, 

 instead of at the extremities of the lower flange, is simply to 

 reduce the effective Jieight or rise from 29.5 ft. to 25 ft. In our 

 example, since the depth of arch is half the whole rise of the 

 centre line, this reduction is considerable. 



For a much longer span and smaller proportional depth the 

 difference would not be so marked, but it would seem that the 

 strains in the second case must always be greater than in the 

 first. The best construction, then, seems to require the hinge 

 in the upper flange at the crown, and at the extremities of the 

 lower flange at the abutments. By this means, the greatest ef- 

 fective rise is obtained, and both ribs aid in supporting the 

 load. Were the hinges all three in the same rib, then, for 

 uniform load, that rib alone is the sole supporting member, and 

 is unassisted by the other. This should then be avoided. 



22. Arcb continuous at Crown, and hinged at Ends of 

 Lower Rib* For this case, referring to Art. 159, we have sim- 

 ply to interpolate from our table there given the values of A, 

 B and y in the equation, 



_ 1 + B K 



y -r^AT K y > 



and thus plot the curve cdeiJc, Fig. 91. The construction of 

 the reactions and horizontal thrust for each weight is then easy. 

 These once known, we proceed as above, in order to find the 

 strains. 



Now, in the f ormnlsB above K 5, and since we can put 



A r 2 



for the square of the radius of gyration, and this radius is 



approximately the half depth of the arch, we have 



25 25 1 



~ (170.6) 2 "" 2910436 ~" 1164" 



Now B and A are, as we see from the table, small, and hence 

 in our present example the terms containing K can be disre- 



