ARCHES AND ARCHED RIBS 177 



Hence the total work of deformation for the entire arch is 



Let b denote the thickness of the arch ring, and consider a section of 

 unit width. Then F=b and / = > and substituting these values 



in the above equation and assuming that E is constant throughout 

 the arch, j .# 12 j 



In Article 139 it was shown that three conditions are necessary 

 for the determination of the linear arch. Therefore, since the values 

 of R and M in the above expression depend upon the position of the 

 linear arch, in order to apply Castigliano's theorem to the integral, 

 /,' and M must first be expressed in terms of these three unknown 

 quantities. which may be conveniently chosen as the position, amount, 

 and direction of the joint pressure at a certain point. 



Having expressed R and M in this way, Castigliano's theorem is 

 applied by differentiating W partially with respect to each of the three 

 unknowns, and equating these three partial derivatives to zero. In 

 this way three simultaneous equations are obtained which may be 

 M >1 vfd for the three unknown quantities, thus completely determining 

 the linear arch. 



The principle of least work, therefore, permits of a rigorously cor- 



rect determination of the linear arch. Instead, however, of actually 



ing out the process outlined above, Winkler has applied the prin- 



ciple to the derivation of a simple criterion for stability, as explained 



in the following article. 



144. Winkler 's criterion for stability. From the preceding article, 

 the total work of deformation for the whole arch is given by the 

 expression 



in which the integral is to be extended over the entire length of the 

 arch. As the position of the pressure line is altered, the first term 

 in this integral changes but little, whereas the second term under- 

 goes a considerable variation, since M = Re, where c is the distance 



