312 



SUPPLEMENT TO CHAP. XIV. 



[CHAP, TX 



reaches the crown, it becomes zero. This point, which gives M , the mo- 

 ment at crown, a positive maximum, is at a distance z = a + a Vf~= 

 0.264911 a, or nearly \ a from the crown. 



The values of V,, H and M a (M 2 and V? being always the moment and 

 reaction at unloaded end), for various values of z, are given in the following 

 table : 



It will be seen that the moment at the unloaded end, -which, as long as 

 the load is left of crown, is the moment at crown also ; increases as the 

 load passes on, is positive and increases up to about a = .25 a. Then it 

 diminishes, becomes zero when the load reaches the crown, changes to 

 negative as the load passes the crown, and this negative value increases up 

 to full load when it is i p a*. For full load, then, the lower end flanges 

 are in tension. At the crown the moment is zero, and the compression 

 there in both flanges is due to H only. 



34. Circular Arch Formulae for V, H and 11. 



(a) Vertical Reaction. 



Here we have r sin /3 = x, r cos /3 d = d x, and P = p dx = pr cos ft d 8. 

 Inserting this in place of P in eq. (43), Art. 18, and integrating betwee/i 

 the limits = a and /3 indeterminate, we have for the reaction at wwloaded 

 end, or for reaction at crown when load does not extend past the crown, 



V = 1^ - I cos a a sin /3 + cos /9 -f- 8 sin 



2 (a Sin a COS a) I- 3 



4- sin a cos a sin |3 |-^- cos a sin* /3 I .... (79) 



8 J . 



