24 Proceedings of the Royal Irish Academy. 



of tlie qwadiaut, where it is nil, then hoop thiiist. There ean be no normal 

 component load at = 45^. The state of stress at B" or 30" from south is 

 given by the pair of nnlike principal stresses j> = 'oi^ and q = -7?^, advanced 

 to tonch the back of tbe rib, one from each of the two load areas, and there 

 compounded by Eankine's ellipse of stress, thns giving an outward oblique 



load of ftp at ^ at east by north. At-4'itis^ south, soon it is east, still out- 

 ward. At = 45 it is wholly tangential, then due north, then west, and so at S' 

 it is again — and normal, but inwai-d. 'ITie whole load is twoequal and 

 opposite couples taken in pairs of opposite quadrants. 



The Balanced Vertical Circular Rib. 



Ifow we are eonsideiing the semicircular masoniy arched bridge, its 

 stability developed from the linear rib. Only a small part of the vertical 

 load is uniform along the span, the superstructure up to tbe level of the rails 

 and the live load, a row of locomotives covering or half-covering the span. 

 By far the greatest part of the load is the weight of the arcb-ring, generally 

 uniform along the curve, or assumed to be so in the first instance. 1 1 cannot be 

 pictured by an area spread on a horizontal stiuight platform, as the heights 

 at the ends reacb up indefinitely. We have devised a simpler way. Again, 

 there is what Bankine caUs the spandril-area, mapped out by a quadrant of 

 the linear arch, and tangents from its crown and springing point. For this 

 load alone there is no normal load at the crown, so that the horizontal hoop- 

 thrust at tbe crown is nil, and so the shape of the corresponding hoiizontal 

 load area is necessarily like a figure-eight, witb the lower positive thrust- 

 area equal to the upper negative pull-area. When built upon a vertical 

 straight base equal to the rise (radius) of the linear rib, the locus (q) crosses 

 the base. The point on the rib, at the height of the crossing-point, is 

 BanMne's point of rapture, or joint of rupture; and its slope, the angle of 

 rupture, he finds to be ^ = 56° = limiting value of ft His construction to 

 build the area fails to give any idea of its correct shape. The horizontal 

 npper outward part of this load cannot be practically applied to masoniy 

 arebes. Its absence compels the linear circular rib in the midst of the 

 upper segmental, elastic, and important part of the arch to become a modified 

 catenary. Now, as the total depth of the load at the crown is always, in 

 railway bridges, a much .smaller fraction of the radius than 1 : 3, the line of 

 stress is a two-nosid cilenari/, a basket-handle-shaped curve. The confining 

 of this altered linear rib within the middle-third of the masoniy arch ring 

 was tbe subject of our foi-mer paper. When this is secured the dejftlis of the 



