CHAP. XIV.] THE BRACED AECH. 2t)l 



analytical formulae for the horizontal thrust and vertical reac- 

 tion at A. Thus, for the vertical reaction, we have, as before, 

 simply 



For the horizontal thrust, however, we have the following 

 very clumsy formula : 



v _ p sin 8 a sin 8 fi + 2 cos (cos ft cos ) 2 (1+ic) cos a ( sin a ft sin /3) 

 2 [a 3 sin a cos a + 2 (1 -t- *) a cos 2 a] 



For the semi-circle, this reduces to 



cog2 



H P 



IT 



K being, as before, = -j -# ; where A is the area and I the 



moment of inertia of the cross -section, r the radius of the arch, 

 and the angles a and /3, as represented in Fisj. 91, viz., the 

 angle of the half span, and the angle to the load, subtended by 

 x. The first of the above formulae is sufficiently simple, and 

 by it we may check the accuracy of our construction. Thus 

 having plotted the curve cdeik by the aid of our expression 

 for y and the tables above for any position of P required, we 

 draw d A d B, and resolve P along these lines, thus finding V 

 and H [Fig. 91]. We can then calculate V from the formulae 



p 



above, viz., V = (a + x). If this calculated value agrees 



2 a 



with that found by diagram, we may have confidence that the 

 curve is properly plotted, and hence that the value of H is also 

 correct. Thus, with very little calculation and great ease, 

 rapidity and accuracy, we can find the reactions at the end A 

 for any given position of P in t any given case. These reactions 

 once known, we can easily find the strains either by diagram, 

 as illustrated in Chap. 1., or by calculation by the method of 

 moments of Art. 14. 



16O. Arch fixed at Abutments continuous at Crown. 

 This is by far the most important case of braced arch, as by the 

 continuity of the crown and fixity of ends we obtain all the 

 advantage possible due to the combined strength and elasticity 

 of the arch. It is also the most difficult case of solution, as the 

 formulae obtained by a mathematical investigation are complex, 



