Arched Ribs of Uniform Section, 389 



SU, the increment of work done ; therefore we have also 



whence the above equations follow. 



In particular -ffr> —rrr> -ran- are the displacements of the 

 r db dti dm Q r 



initial section from its original position produced by the action 



of the load on the rib. Thus in the case of the chain of circular 



1 /7TT 

 links, ^ • 77^- is the stretching of each link ; and when multi- 



Jcil clr 



plied by the number of links, would give the stretching of the 

 whole chain by the action of the weight W. 

 March 1865. 



NOTE. 



This article is intended chiefly to illustrate the application of the 

 principle of least action, many details are therefore omitted which 

 are necessary to complete a practical solution of the problem. Also 

 the simplest cases have been chosen, to avoid, as far as possible, the 

 inherent complexity of these problems. 



If a chain hang beneath the rib and carry part of the load by ver- 

 tical struts, then, the form of the chain being known, the load on each 

 strut is known in terms of H' the unknown horizontal tension of 

 the chain ; for otherwise the form of the chain would not be pre- 

 served. Supposing the struts indefinitely many in number, the in- 

 tensity of the load on the chain is H' — % ; -^ being got from the 



dx dx 



equation to the chain (x horizontal). 



The intensity of the vertical load on the rib is consequently less- 

 ened by the quantity H' — ^, and the coefficients , , 



dx 2 dH dF dM 



must be calculated, together with a new coefficient — , in terms 



dH-'o 

 of H' and the other quantities involved. U can now be made a 

 minimum, and the four constants F , H , M , H' determined, 

 whereby the problem will be fully solved. If the chain take the 

 thrust of the rib, then H = H' , and there are only three coefficients. 

 The general problem of the stiffened suspension bridge is a particular 

 case of this more general problem. The elasticity of the chains can 

 be taken into account by estimating the work done in them in terms 

 of H' . 



Many other problems of the same general class, that is, where the 

 law of variation of stress is known, and its absolute amount is re- 

 quired, may be advantageously treated by application of the principle 

 of least action ; in a future article I hope to consider some cases in 

 which the law of variation is required. 



