268 THE BKACKD AEOH. [CHAP. XIV. 



strain it tends to cause in the piece in question. Properly, 

 since when H is below Cj is negative, we should have H (\ 

 for moment causing compression in C. 



Thus we may proceed till we pass P, and then the moment 

 of P, with its proper sign, as producing tension or compression 

 in the piece in question, must also be taken into account, or we 

 may instead take the moments of V and H at the other end, 

 that is, the same side of the weight as the piece itself. 



The diagonals may be similarly found by moments. It will, 

 however, be best to determine them by diagram, one of the 

 flanges being first calculated (in this case the first upper flange), 

 as explained in Art. 125. They may also be calculated from 

 the resultant shear at any apex. Thus, for diagonal 3 find the 

 vertical components of the previously determined strains in D 

 and C. These vertical components, together with the vertical 

 component of the strain in 3, must for equilibrium be equal and 

 opposite to the total shear at b. 



Calling this shear F, and a, y3 and y the inclinations of D 

 and 3, we have for the strain in 3, 



S 3 = (F S x sin a S 2 sin /S) cos 7. 



If either of the vertical components of the strains in D or C 

 acts opposite to the shear F, it must, of course, be subtracted ; if 

 in the same direction, added to F. For the ready determina- 

 tion of the proper signs, see Appendix, Art 16 (4). 



The moment H d is the moment at the fixed end, and is con- 

 stant throughout the arch for any one position of the load. It 

 causes tension in outer and compression in inner flanges, pro- 

 vided, as in the Fig., <f> fall below the centre of the end section. 

 This moment is increased (or diminished if < is above) by the 

 varying moment of H for each apex. 



The above method of determining the strains in the braced 

 arch, though not strictly graphical, but rather a combination of 

 analytical and graphical methods, offers such a ready solution 

 of this important and difficult case, that we have not thought it 

 out of place to notice it somewhat in detail. We consider it by 

 far the simplest and easiest method which has yet appeared. 



163. Analytical Formulae for V and H. A comparison 

 of our method with the long and involved analytical expres- 

 sions to which the theory of flexure conducts us, will render its 

 advantages still more apparent. 



