Notes on Iron Ardies. 85 



however, show that this is not the case, for if arches be 

 employed' it will be necessary to have a distinct trough or 

 tube, separate from but supported by the ribs, whereas if the 

 girder principle be adopted the girders themselves may be 

 made to assume the form of a trough or tube, thus dispensing 

 with any separate structure to contain the water ; and in this 

 way the balance will be turned against the arch in the 

 question of economy of material. 



Let us now endeavour briefly to analyse the stresses 

 endured by the material of an arched rib under varying con- 

 ditions of temperature, load, &c. 



We will first assume that the arch as originally designed 

 ^ is of a form adapted to the dead or unvarying load to be 

 boiTie, which form in the usual case of a uniformly distri- 

 buted load is a parabola having its axis vertical ; and it may 

 further be remarked that a circular curve will usually be 

 found not to deviate in any important degree from the 

 parabola, and is, from a practical point of view, decidedly pre- 

 ferable. Let us also assume that the rib is hinged at the crown 

 as well as the springing. Let W represent the total weight 

 of the structure, which may usually be taken as uniformly 

 distributed over the whole length of the rib, h the span and 

 h the rise of the arch ; then the compression of the rib will be 



-i— at the crown, and at every other point — -; — '—^ when 9 

 8h 8h 



is the angle made by a tangent to the rib at the point in 



question with a horizontal line, and this compression will be 



uniformly distributed over the whole cross-section of the 



rib in every case. In other words, there will be no approach 



to a cross-bending action on any part of the rib, even though 



the temperature should vary or the abutments yield slightly 



to the thrust of the arch. If an additional load of W, 



uniformly distributed, be placed upon the bridge, these 



•n T. (W+W) I , (W+W') I sec. 

 compressions will become '^ —^ — ^— and ^^ ' i- - 



respectively, and the perfect freedom from cross-bending 

 before mentioned will still be maintained. If, however, the 

 live load, instead of being uniformly distributed over the 

 whole span, cover a part of it only, a cross-bending action 

 will come into play, which will attain its maximum when 

 half the bridge is loaded, and which will be unimportant or 

 severe according as the Live load is small or large compared 

 with the weight of the structure. The tendency of this 

 cross-bending action wiU be to increase the radius of curva- 



m2 



