THE BRACED ARCH. 



[CHAP. xiv. 



of intersection lines to A and B, and resolving P in these direc- 

 tions. 



The equation of this locus can be found analytically without 

 much difficulty. 



\st. PARABOLIC ARC. Thus, for a parabolic cwc, we have* 



32 a 2 A 

 y ~ 5 (5 a 2 - a?}' 



Where [PI. 23, Fig. 91] a is the half span, and h the rise of 

 the arc ; x the distance of the weight from the crown, and y 

 the ordinate N 6? of the locus cd eik. 



For a given arc, then that is, a and h given we have only to 

 substitute different values for a;, as x = 0, 0.1, 0.2, etc., of the 

 span, and -we can easily find the corresponding ordinates y, and 

 thus construct the locus cdeik. It is then easy to find the 

 reactions at A and B for any position of P, as above indicated. 



The vertical reaction at the abutment may also be easily 



found by moments thus, 



p 

 V x x 2 a = P (a + ), or V t = (a + x). 



2 a 



The horizontal thrust is 



_ - 



~ 64 



a 8 A 



These values, though not needed for the construction above, 

 may be of use, and are therefore given. In the following 

 tables we give the values of H and y for different values of a? : 



* For the demonstration of the analytical results made uee of in this chap- 

 ter, we refer the reader to Die Lehre von der Elasticitlit und Festigkeit, by B. 

 Winkler. Frag, 1867. See also the Supplement to this chapter. 



