326 THE STONE ARCH. [CHAP. XV 



rirntly great to unite the whole abutment as a single block 

 which turns about its under edge, gives too small dimensions. 

 To ensure safety it is assumed that equilibrium exists with 

 reference to rotation about the lower edge, when the thrust of 

 the arch is 1.5 greater than the actual. Investigations of 

 French engineers have shown that this coefficient of safety for 

 very light arches is not less than 1.4. The old tables of Petit 

 give 1.9. We assume it, therefore, = 2. 



If, therefore, the double thrust of the arch at the lower point 

 of rupture is united with the weight of the abutment, the re- 

 sultant should still fall within the base. Since it is indifferent 

 in what order the elements of the abutment are resolved, it is 

 best to divide it into vertical laminae, and unite these with the 

 double thrust. The equilibrium polygon thus obtained should 

 cut the foundation base within the edge of the abutment. 



When the thickness of the abutment is thus determined, we 

 must construct the actual pressure line for the simple thrust 

 in order to determine the joints. In drawing this second pres- 

 sure line, we should properly take the divisions of the arch by 

 the joints themselves. If, however, we take the division in 

 vertical laminae, the deviation, as we have seen, is insignificant. 

 The normals to the actual joints must, then, not deviate from 

 the sides of this pressure line by more than the angle of 

 repose. 



178. Construction of the Pressure Line. In PL 25, Fig. 

 105, we give the method of construction of the proper width of 

 abutment for an arch. We first divide the arch into vertical 

 laminae, and determine their weight. If the surcharge has 

 vacant spaces, or is generally of different specific weight from 

 the material of the arch itself, it must first be reduced. Thus, 

 if the surcharge (spandrel filling, etc.) weighs, for instance, only 

 f ds as much as an equal area of masonry in the arch, we have 

 simply to diminish the vertical height above the arch by d. 

 We thus obtain the dotted line given in the Fig., which forms 

 the limit of the reduced laminae, and we can treat the areas 

 bounded by this line the vertical lines of division and the 

 intrados as homogeneous. We have then only to determine 

 the centres of gravity of the various laminae according to the 

 construction for finding the centre of gravity of a trapezoid 

 (Art. 33), and suppose at these points the weights, which are 

 proportional to the reduced areas of the trapezoids to act. 



