1847.] 



THE CIVIL ENGINEEERAND ARCHITECT'S JOURNAL. 



67 



There are three conditions to be considered as affecting the equili- 

 brium of tlie arch — 1st. The form and dimensions of the inlrados or 

 internal lower curve of the arcli : 2nd. The form and dimensions of the 

 extrados or external curve of the arch: 3rd. The weight and dis- 

 position of the loading. The first condition of courses includes the 

 rise and span ; the second, combined with the first, is equivalent to a 

 determination of the depth of the voussoirs; the third exhibits the 

 fxternal forces to which the system is subjected : for we suppose that 

 not only is the weight of the loading known, but also the manner of 

 its distribution — that is, whether and in what degree its specific 

 weight or density varies in different parts of it. 



But, it may be asked, is not this enumeration of the conditions of 

 equilibrium imperfect from the omission of the form, number, and 

 position of the joints? The answer to this question is important, be- 

 cause it exhibits in the strongest light possible the distinction between 

 the ancient and the modern theory of the arch. It is manifest that 

 the extrados and intrados, which have been enumerated as two of the 

 "conditions," simply define the depth of each voussoir and the form 

 of its upper and lower curved surfaces: the lateral dimensions of the 

 voussoir, and the form of those surfaces of it which are in contact with 

 the adjacent voussoirs, are as yet left indeterminate. Consequently 

 it is not known how many joints there be, or even whether there be 

 any joints. Neither is the direction of the joints ascertained ; they 

 may be plane converging joints, or they may be "joggled," or all 

 vertical, or all horizontal — (as an instance of the latter case maybe 

 cited the Treasury of Atreus at Mycene, where the stones are not 

 wedged together, but are merely disposed in horizontal courses, each 

 resting upon and over-lapping that beneath it, the whole being heven 

 in such a form as to give the structure the appearance, but not the 

 mechanical properties, of a dome). 



It is as well to consider these objections in limine. The answer 

 then is this. The theory which Professor Moseley lias exclusively, 

 and Professor Whewell and other writers partially, adopted, and 

 which we are endeavouring to set before the reader in a new form, 

 presupposes that the stones are sufficiently cuneiform to be incapable 

 of sliding past each other: but further than this no consideration is 

 paid to the form of the bed-surfaces. The mutual friction of the 

 voussoirs is so great, that there is not the least difficulty in so shaping 

 them, that the arch shall not fail by their sliding past each other — 

 that accident, as has been said, is never known to occur. So that the 

 precautions under this head being perfectly obvious, the theory does 

 not at all deal with them, and, as we shall see further on, is independent 

 of the form and direction of the joints — or to speak more precisely, the 

 theory shows how to ensure the stability of the arch, whatever may be 

 the inclination of the beds, ic, however numerous, or however few 

 they may be ; so that the structure should stand even if intersected by 

 infinite du mberof joints running in every possible direction — provided 

 always that they were sufficiently convergent or joggled, so that the 

 voussoirs could only fail, if they did fail, by opening, and not by 

 slipping. 



The reader who approaches the subject for the first time will not 

 perhaps perceive at once the full effect of these observations. But his 

 attention is now directed to them, as he will be left hereafter to apply 

 them for himself to the cases particularly discussed. Returning now 

 to the three " conditions" which have been enumerated, we find that 

 they give rise to four distinct problems — three of the problems arise 

 from combining any two of the conditions as data to find the third, 

 and the fourth problem arises from the combination of all three con- 

 ditions considered as known data. The four problems are these — 



1. Given the distribution of the loading and the intrados to deter- 

 mine the extrados proper for stability. 



2. Given the distribution of the loading and the extrados to deter- 

 mine the inlrados proper for stability. 



3. Given the extrados and intrados to determine the distribution of 

 the loading proper for stability. 



4. Given the extrados, the intrados, and the amount as well as dis- 



tribution of the loading to find the consequent horizontal thrust of the 

 arch. 



It will be observed that the absolute weight of the loading is taken 

 into account in the fourth problem only : in the other problems the 

 relative weight is alone considered ; in other words, the form of the 

 arch is connected not with the actual amount of the external forces, but 

 with the relation or comparative amount of those forces, as they vary 

 in different parts of the structure. It may perliaps be as well to re- 

 collect that in this respect there is an analogy between the arch and 

 the other two contrivances employed for spanning the interval be« 

 tween piers or abutments— namely, the catenary and the girder. For 

 respecting the catenary, it is known that if a heavy cable and a fine 

 thread, both of the same length, span the same distance, they 

 will assume the same form : and this similarity of form will be ob- 

 served either when the cable and the thread are each of uniform 

 thickness throughout its length, or when (the thickness not being uni- 

 form) the law of variation is the same for each. Also the form of the 

 girder of uniform strength, or the variation of the sectional dimen- 

 sions of a girder that it may be equally strong in every part, depends 

 not on the actual amount of the load, but on its distribution or com- 

 parative weight in different parts. This digression will not appear 

 superfluous when it is considered how much the physical conception 

 of a subject like the present is cleared by a distinct apprehension of 

 its relations to kindred subjects. 



We shall take the fourth problem first, because it is the simplest. 

 Here, however, as elsewhere, it is simply intended to show how much 

 may be done by very simple calculations — the discussion of the pro- 

 blem in all its generality, and with perfect accuracy, vpill not be at- 

 tempted. 



Thrust of jirchea. 



The thrust or horizontal pressure of the arch tending to push its 

 abutments outwards is the distinguishing charticteristic of that struc- 

 ture. This kind of force does not exist in the other contrivances for 

 spanning the intervals between abutments. The suspension chain, 

 attached at its extremities to the summits of two towers, exerts at 

 those points a horizontal force inwards, equal to the tension of the 

 catemry at its lowest point. In the beam or girder, the forces of 

 tension and compression are equal and opposite, and there is, there- 

 fore, no external horizontal force on the abutments. 



The following method will explain the cause of the horizontal 

 thrust of the arch, and serve in estimating its amount. Let the ac- 

 companying diagram represent the half of an arch, and its own load 

 considered as a separate statical system. And to make the case sim- 

 pler, let us suppose the load on the half arch sustained entirely by it, 

 and not supported in any way by the contiguous portions of loading 

 on either side of it, which are supposed to be removed. The hall- 



load here represented will not, in that case, have any external force 

 actii g upon it to modify the effect of its weight on the half arch. 



:iu» 



