CHAP. XIV.] THE BRACED ARCH. 267 



162. Illustration of Method of Solution. As an illustra- 

 tion, take a portion of a braced arch, as represented in PI. 24, 

 Fig. 94. We have first to plot the upper curve or locus of m, 

 for the given dimensions of the centre line of the arch. This 

 curve once plotted, then, for any position of the weight, we have 

 only to prolong P to ra, and draw a line from -m to the end of 

 centre line if the arch is hinged at ends, or to </> at a distance 

 GIJ above or below the end of centre line if the arch is fixed at 

 ends ; c t being easily found from our formulae above. In sim- 

 ilar manner, we draw a line from m to the other end, or (%. 

 Now these two lines are the resultants of the outer forces P, 

 and by simply resolving P in these directions, we have at once 

 V and H, while the moment at the end M x = H c 1? positive 

 if it tends to cause compression in lower flange, or since GI is 

 negative down, if it acts below the end. 



We can now easily find the strain in any flange, as P, 

 whether the arch vary in depth or not, provided only it is sym- 

 metrical with respect to its centre line. Thus for D, take the 

 opposite apex a as the centre of moments. The moment of H 

 with reference to #, as shown in the Fig., tends to cause tension 

 in D, while that of V causes compression. We have then, 

 representing tension by minus, 



. . _ moment of V 

 strain m D = 



lever arm of D 



all with reference to a. If the result is minus, it indicates thus 

 tension, if plus, compression ; if it is zero, the two moments are 

 equal, and at a, therefore, no moment exists ; hence a must be 

 a point of inflection. Note that H and V must be taken as act- 

 ing at <, Fig. 94. We can also evidently take them as acting 

 at the centre of the end cross-section, if we take into account 

 the moment H c x . 



In similar manner, for C we take b as centre of moments, and 

 then, since H now causes compression in C and V tension, we 

 have for V and H, acting at <, 



moment of H moment of V 



strain in C = -= . 



lever arm of C 



9 

 For V and H, considered as acting at the end of centre line, we 



, _ moment of H -f H c, moment of V 



have C = - , 



lever arm or 



taking <\ without regard to its sign, but simply to the kind of 



