270 TOE BRACED ARCH. [CHAP. XTV. 



where K = -j 5 ; I being the moment of inertia, and A area of 



the cross-section, and r the radius of circle. 



These formulae, it will be observed, involve much labor in 

 any particular case. Where the number of weights is large, 

 the computation is tedious in the extreme, A method which 

 shall give accurate results and avoid such formulae as the above 

 is certainly very desirable, and such we believe to be the 

 method which we have given. 



For the analytical investigation of arches, and the demon- 

 stration of the formulae for the curves of which we have made 

 use, the reader may consult Die Lehre von, der Elasticitaet und 

 festigkeit, by Dr. JK. Winkler, to which we have already re- 

 ferred, and which contains a thorough discussion of the whole 

 subject. The tables which we have given, as well as the for- 

 mulae for y, G! and c& will, it is hoped, give the method here 

 presented a practical value, and render the solution of any par- 

 ticular case easy and rapid. 



164. For a solid or plate girder arch of given cross-section, 

 we may also determine the proper proportions by finding, as 

 above, the moment M of the exterior forces at any point. 



TI 



Then M = , 



where T is the strain per unit of area in any fibre distant t 

 from the axis, and I the moment of inertia of the cross-section. 

 Thus, for a rectangular cross-section I = y 1 ^ b d s , where b is the 

 breadth and d the depth. 

 Hence M = -J- T b d?, 



if we take t = -. 

 2 



The strain, then, per unit of outer fibre will be 



T _ 6M 



= Jd?' 



The safe working strain should not exceed for iron 5 tons 

 per sq. inch for tension and 4: tons for compression, and there- 

 fore d being assumed, we can easily proportion b so as to sat- 

 isfy this condition. 



As examples of braced arches, such as we have considered, 

 viz., continuous at crown and fixed at abutments, we may men- 



