ORAP. IV.] SUPPLEMENT TO CHAP. XIV. 291 



CHAPTER IV. 



ARCH FIXED AT ENDS. 



12. Introduction. In the previous case, the end reactions pass 

 always through the ends. If, however, the ends are " walled in," so that 

 the end cross-sections remain unchanged in position, and cannot turn, these 

 reactions pass then no longer through the centres of the end cross-sections. 

 In the first case, the moments at the ends are zero ; now, however, we have 

 end moments to be determined, viz., Mj and M 2 , left and right. For their 

 determination we have the condition that the tangents to the curve at the 

 ends must always remain invariable in direction, or for the ends, A $ = 0. 



In the arch above with hinges at ends, we have always considered a por- 

 tion lying between the end and any point. In the present case, however, 

 we shall consider the portion between the crown and any point. Both 

 methods lead, of course, to the same results, but the latter, in the present 

 case, is somewhat simpler. 



Accordingly, we conceive the arch cut through at the crown [PL 24, 

 Fig. 93]. The total resultant force exerted upon the one-half by the other, 

 we decompose into a vertical force V at the crown, and a horizontal force 

 H. The distance c Tc of this last from the centre of gravity of the section 

 at crown is e t , and hence the moment at crown is M = H e a . 



13. Concentrated Load General Formulae. Let a weight 

 P act at any point; then representing, as before, by primes, quantities 

 relating to the portion between the load and right end, we have, as in (18), 



G = H cos <f> (P V) sin 0, G' = H cos < + V sin $ ) 

 N = H sin $ + (P V) cos $, N' = H sin < V cos (f> ) " 



Also, M = H (<? + y) Vx + P (x ), M' = H (<? + y) V x, 



or, since H = M = moment at crown, 



M = Mo H y V x + P(x ), M' = M H y V x . . (32) 



(a) Intersection curve. 



The two reactions, R and R', intersect, as before, in a point L (Fig. 92), 

 which must lie upon P prolonged, as otherwise R, R' and P could not be 

 in equilibrium. The locus of the point L we call, as before, the intersection 

 curve. The equation of this curve can be easily found when V, H and M 8 

 are known. 



The force acting upon the portion B E (Fig. 92) is the resultant of V 

 and H. The component H acts at the point of intersection o of this 

 resultant L ^ with the vertical through O. The vertical distance of this 

 point o from c is, as above, e<> ; its horizontal distance from P is z. Then 

 z cot L ^ Ic is the vertical distance of this point from L, and o + * cot 



