20 Proceedings of the Royal Irish Academy. 



his criticism, that tlie catenary method neglected horizontal loads, we made 

 a graphical development of Eankine's conjugate loads, printed for private 

 iise in the Engineering School of Trinity College, Dublin. In this we were 

 allowed by the Academy the use of the blocks and part of the scientific work 

 of our paper. 



In this paper we will give our latest development of conjugate load areas, 

 and use the blocks of the privately printed paper, but only to aid the 

 scientific discussion. Still these blocks have sufficient information on their 

 faces to express their practical use to the engineer. 



Hankine's SUreostatic Arch. 



This refers to a linear rib, generally liaving two quadrants, which he calls 

 a complete arch, horizontal at the crown and vertical at the springings. It 

 is balanced by two conjugate loads — a vertical load varying in a symmetrical 

 definite way horizontally outward from the crown in each direction, and 

 resisted as a whole by the two upward supports at the springings, and 

 also bearing a pair of horizontal loads, which are distributed in a definite 

 vertical way so as to prevent the arch from spreading or collapsing at any 

 point ; so that if the rib be supposed to have a plenum of hinges, there is no 

 bending, but only thrust (or tension), at each point due to the total load. 

 Eankine's theorem is, that of the three things — the shape of the curve, the 

 distribution of the vertical load, the distribution of the horizontal load ; 

 given any two, the third is determinate. 



We shall deal piincipally with the semicircular rib, and suppose the 

 vertical load spread in some definite manner on a horizontal platform equal 

 to the span ; these two given, we shall find the necessary distribution of the 

 horizontal load upon two vertical platforms equal to tlie rise of the rib. In 

 this diagrammatic way we have two sets of conjugate load areas. 



The Balanced Cimdar Horizontal Rib. 



For the circular rib, fig. la, the given north-south load areas are a pair 

 of parallelograms representing a uniform stress ^j, spread on the barricades, 

 of which «6 is one base for the north-west quadrant. They tend to make 

 the rib collapse. By symmetry the conjugate load areas, east-west, that 

 resist the collapse is the uniform stress, equal and like, q = p, in part shown 

 on the base (or barricade) ccl. These Eankine calls equal like principal 

 stresses, or, more shortly, a fluid stress, and this correctly describes the state of 

 stress of the fluid surrounding the cylinder, or of the granular mass of 

 earth in level layers surrounding it, and means that cubes of the surrounding 



