SUPPLEMENT TO CHAP. XIV. [CHAP. VT 



For a = 90, or for aemi-circk, we have from (80) 



37r-6/3+18sin/3cos/3-12sin*+6rBin/3 

 L 



4- K (-8 JT 12 /3 sin* +6 -6 sin j3 cos /3+6 ;r sin /3+2 TT sin 3 |. 

 For /3 = 90, or for full load upon semi-circle, 



pr [6,-8.]. 



24 



7-*] 



(c) Moment at unloaded end. 



From Art. 18 (c) we have, for concentrated load, 



/_ a \ . (sin a sin /9) f 



I 2 cos a : I t P r : ^-. 



\ sin nj sin a 



The value of H we have already given in (80). 



Inserting in the second term p d x = p r cos ft d & for P, and then inte- 

 grating, we have 



M = AH-B ....... (82) 



where A = i r [ 2 cos a ^ ] = - (2 cos a ^ 

 y sin a/ 2 sin a y sin 



v a* / 



and B = - . ( sin a si 



12 sin 3 a \ 



For a uniform load over whole span, /3 = a, and 



/- o \ 2 pr 3 sin 2 a 



M = Hr(2-cos-^- .. . (83) 



y sm a/ 3 



We have from (83), by series, the approximate formula for moment at 

 crown 



where A is the area and I the moment of inertia of cross-section, g the 

 radius of gyration, or, approximately, the half depth for framed arch as 

 always a negative moment indicates tension in lower or inner flange. 



Equations (79), (80) and (82) may, if desired, be tabulated as in Art. 18. 

 For small central angles, or for h small with respect to a, K may be disre- 

 garded, and the results already given for parabola (Art. 33) may be taken 

 as sufficiently exact. 



