180 STRENGTH OF MATERIALS 



the case of an arch intended to support a roadway, the level of which 

 is fixed, the loading depends to a large extent on the form of tin* 

 arch, and consequently the equilibrium polygon cannot be determined 

 until the form of the arch has been assumed. 



In designing arches, therefore, the method usually followed is to 

 assume the form of the intrados of the required arch, and determine 

 its thickness at the crown by an empirical formula, such as those 

 given in the preceding article. Then, having drawn ihc extrados and 

 load line, the surface between the intrados and the load line is 

 divided into any convenient number of parts by drawing verticals, 

 and the amount and position of the resultant weight of cadi part iW 

 a section one foot wide is calculated. An equilibrium polygon t>i 

 this system of loads is then passed through the middle point <>f the 

 arch ring at crown and abutments by the method given in Article 1 ."> 4. 

 If this equilibrium polygon lies within the middle third uf tin- arch 

 ring, the arch is assumed to be stable against overturning. 



If the equilibrium polygon through the middle points of tin- ar<-h 

 ring at crown and abutments does not lie entirely within the mid- 

 dle third of the arch ring, these three points are shifted so as 

 to make it do so if possible. If no choice of the tlnv- ]M,im> will 

 make the equilibrium polygon lie entirely within the middle third 

 of the arch ring, the design must be altered until this has been 

 accomplished. 



The next step is to calculate the maximum unit joint pressure by 

 the formula given in Article 141, and apply the criterion f<>r stability 

 against crushing given in Article 140. When these criteria have been 

 satisfied the design is assumed to be safe. If, however, there is a 

 considerable excess of strength, the design may be lightened and the 

 criteria reapplied. 



Before the design can be considered complete' it must also be 

 shown that the above criteria are satisfied for every form of loading 

 to which the arch is likely to be subjected. In the case of an an-h 

 designed to carry a heavy live load, such as that due to several 

 locomotives, it may be necessary to draw a number of load lines 

 corresponding to different positions of the load, and make a cor re- 

 spending number of determinations of the equilibrium polygon and 

 maximum joint pressure. 



