ARCHES AND ARCHED RIBS 179 



In order for an arch to be stable at least one of the many possible 

 assumptions of the linear arch must be such as to fall within the 

 middle third of the arch ring. Moreover, the elastic deformation of 

 the arch is such as to move the linear arch as near to the center line 

 as the form of the arch permits. Therefore, if for any given arch 

 it is possible to draw an equilibrium polygon which shall everywhere 

 lie within the middle third of the arch ring, the stability of the arch 

 is assured. 



This criterion for stability is due to Winkler, and was first given 

 l.y him in 1879. 



145. Empirical formulas. The thickness necessary to give an arch 

 at the crown can only be found by assuming a certain thickness and 

 determining whether or not this satisfies all the conditions of sta- 

 bility. The least thickness consistent with stability is such that the 

 average compressive stress does not exceed one twentieth of the 

 ultimate compressive strength of the material. The arch is usually 

 made somewhat thicker than is required by this criterion, however, 

 for the thicker the arch the more easily can the equilibrium polygon 

 be made to lie within the middle third of the arch ring. 



The following empirical formulas for thickness at crown represent 

 the best American, Knglish, and French practice respectively, and 

 may be used in making a first assumption as a basis for calculations. 



-I- O.2 : Trautwine. 



r = radius of intrados in feet ; d = rise in feet ; 



/ = span in feet ; b = depth at crown in feet. 



146. Designing of arches. In designing an arch to support a given 

 loading the equilibrium polygon for the given system of loads should, 

 in accordance with Winkler's criterion, be assumed as the center line 

 of the arch. This, however, is not always possible. For instance, in 



