38 JOURNAL OF THE 



Analytical theories of the solid arch have been devel- 

 oped by Winkler, Greene and others, and the graphical soln- 

 tion has been given by Prof. H. T. Eddy, to which the 

 writer has contribnted his mite. 



In Vail N'osh'aud' s Ejigineeriiig Magazine for Jannary 

 and November, 1879, the writer claimed that the theory of 

 the solid arch was the most exact solntion of the vonssoir 

 arch, and a graphical treatment was given in the last naijied 

 article. In the same year Castigliano, Winkler and Greene 

 referred the treatment of the vonssoir arch to that of the 

 solid arch, and finally, in 1893, the writer, in the second 

 edition of ''Theory of Vonssoir Arches," has given 

 extended applications of the theory of solid arches to vons- 

 soir arches by two distinct methods, one fonnded on the 

 analytical method in part and the other entirely graphical. 



These methods were independently applied to a nnmber 

 of stone arch bridges, whose rise was one-fifth the span, 

 for a loading known as Cooper's "Class Extra Heavy A," 

 so placed as to prodnce the maximnm departnre of the 

 resistance cnrve from the center line of the arch ring, and 

 the resnlts appear to be of snch importance as to offer some 

 apology for writing this article. 



In the stone arch bridges examined the specific gravity 

 of the voussoirs was assnmed at 140 pounds per cubic foot, 

 and the density of the spandrel backing was taken at eight- 

 tenths that of the voussoirs. The weight of this backing 

 and any loads on the bridge was assumed to be transmitted 

 vertically to the arch. It is true that this may not be 

 exactly true; in fact, the spandrel may act as an arch itself 

 to some extent, still such additional security may be sup- 

 posed to neutralize the dynaiuic effect of moving loads, the 

 static effect of the loads being met by designing the arch 

 ring, supposed of constant section throughout, so that for 

 the most unfavorable position of the load, for <7;2v joint, the 

 line of the centers of pressure on the various joints should 



