256 . THE BRACED ARCH. [CHAP. XIV. 



two pieces which meet there. The strains in these two pieces 

 being thus found, those in two others in equilibrium with each 

 of them may be obtained. In Art. 13 we have already illus- 

 trated the method of procedure for such a case, as also the 

 method of find ing graphically the resultant at crown and abut- 

 ments due to any position of the weight. 



Thus the resultant at the crown for the unloaded half must, 

 for equilibrium, pass through the hinge at B also. Its direc- 

 tion is thus constant for all positions of P upon the other hatf. 

 The resultant for the other half must then pass through a and 

 the hinge at A (Fig. 90). 



We have then simply to draw a B, prolong P to intersection 

 a, and draw a A. A a and B a are the directions of the resul- 

 tant at A and B, and by resolving P along these lines, we may 

 find the vertical reaction V = a b and the horizontal thrust 



n = cl>. 



We can thus easily find the reactions at the abutments in 

 intensity and direction, and following these reactiors through 

 the structure, as illustrated in Arts. 8-13, Chap. I., can deter- 

 mine the strains upon all the pieces for any pontion of the 

 weight. A tabulation of the strains for each weight will then 

 give us the strains for uniform load as well as I've load, as al- 

 ready explained in the preceding chapter, Art. -56. 



There must be only two pieces meeting at the abutments. 

 Thus the pieces in Fig. 90, represented by Iroken lines, can 

 serve only to support a superstructure, or transmit load to the 

 arch, and have no influence upon the strains n the other pieces. 



If the span A B = 2 , the rise of the arh is A, and the dis- 

 tance of the weight P from the crown is x. positive to the left ; 

 then taking moments about the end B, we have 



2 Va = P (a + x), or V = P ^ + X \ 



A (lr 



Similarly, taking moments about the crown, 



Va Pas P (a x) 



-Va + Uh=Px. or H =r 7 = v . , '-. 



fa 2/i 



The same formulae apply for a weght upon the other half, for 

 V and H at the other end. 



The values of V and H can etsily be found from these for- 

 mulae, and the strains then calculated by moments, thus check- 

 ing the diagrams. If these reactions are found for the giveD 



