ARCHES AND ARCHED RIBS 181 



The stability of the abutments still remains to be investigated, and 

 finally the bearing power of the soil on which these abutments rest. 



Problem 162. Design a concrete arch to span a stream 25 ft. in width and sup- 

 port a roadway 15 ft. above the level of the stream, if the spandrel filling is clay 

 weighing 120 lb./ft. 3 ; the maximum depth of frost is 3| ft. and the bearing power 

 of the soil at this depth is 4 tons /ft. 2 (see Article 158). 



147. Stability of abutments. To determine the stability of the 

 abutments, the joint pressure at the haunch is combined with the 

 weight of the abutment into a single resultant, say R 1 . For stability 

 against overturning, the line of action of this resultant must strike 

 within the middle third of the base (Article 140, 2). 



Resolving the resultant R' into a horizontal component R h and a 

 vertical component R v , the maximum pressure on the soil is calcu- 

 lated by substituting this value of R v for R in the formula given in 

 Article 141. To prevent sinking of the abutments, this pressure must 

 not exceed the bearing power of the soil (see Article 158). 



For stability against sliding, the shearing stress between the abut- 

 ment and the soil, due to the horizontal component R h of the result- 

 ant R', must be less than the friction between the two ; or, more 

 briefly, the angle which R' makes with the horizontal must be less 

 than the angle of repose (compare Article 164). 



148. Oblique projection of arch. Suppose that an arch, its load line, 

 and its pressure line are drawn to any given scale, and then the whole 

 figure is projected upon an oblique plane by a system of parallel lines. 

 The projection of the pressure line on this oblique plane will then be 

 the true pressure line for the projected arch and its projected load line. 



This principle can often be used to advantage, as, for example, in 

 comparing two arches of equal span but different rise. Its most 

 important application is in giving an accurate construction of the 

 pressure line for arches of long span and small rise. Thus, instead of 

 plotting such an arch to scale, its projection can be plotted ; or, in 

 other words, its span can be shortened any convenient amount. A 

 larger unit can then be used in plotting the vertical dimensions than 

 would otherwise be possible, and consequently the pressure line can 

 be drawn to any desired degree of accuracy. 



Having constructed the pressure line in this way, the pressure on 

 any joint of the given arch can be found from the pressure on the 



