232 



THE CIVTL ENniNEER AND ARCHITECT'S JOURNAL. 



[July. 



AN ESSAY ON THE CONSTRUCTION OF OBLIQUE 

 ARCHES.* 



By Edwakd Sang, M.S.A., Civil Engineer, Edinburgh. 



(Mridgcdfrom the Edinburgh New Philosophical Journal for April.) 



ScAHCF.LY any hrancli of civil engineering Ijcars so closelv on tlie 

 advancement of civilization as (lie art of road-nialuiig. Tile "immense 

 sums tliat are annually expended on them evince the importance of 

 onr roads. Our object is not merely to find a path from one town to 

 another, we must be transported in the most expeditious manner 

 possible. Is there a declivity ; thousands are spent to remove it: is 

 a road suspected of being a few yards longer than is needed ? a new 

 line is immediately chalked cut. One might almost imagine that a 

 monomania had seized us, and that the tulip, the dog, the pigeon, 

 and ail the other f mciers had deserted their peculiar departments to 

 concentrate their energies on this one grand matter of roads. The 

 madness is ii very reasonable one ; for if there be a hill, multitudes 

 daily climb aud descend it: or if a road be circuitous, the quantity of 

 unnecessary travelling might soon be sufficient to carry one com- 

 fortably round the globe. 



While journeying, we are often annoyed by bridges. Sometimes, 

 for cheapness, they have been erected far out of the line of road, and 

 we enjoy, on one side of a river, the delightful prospect of doubling 

 along the other. At other times, after skirting the banks as if on a 

 journey to the source, we are all at once wheeled right across the 

 water, and ere we are certain that our necks are yet safe, an equally 

 sudden turn restores us to our original direction. And occasionally 

 our vexation is crowned by an altercation between the drivers as to 

 w hich of two veliiclcs is bound to back down the steep slope of some 

 imtiquated erection. That time has now gone by when a bridge of 

 any kind was hailed with satisfaction; we have scarcely such a thing 

 as a ford wdierewith to contrast it, ;uid liaving only bridges to compare 

 with bridges, we have become somewhat nice in our taste. Many of 

 the old high-backed bridges have been replaced by others with level 

 road-ivays; these again by bridges with road-ways inclined to suit the 

 elevation of the opposite banks, and now another improvement is be- 

 ginning to be called for, tliat of crossing the river obliquely, so as to 

 make the bridge harmonize with the general line. This we may con- 

 sider as the ve plus ultra in bridge building, for then the road-way 

 over the bridge coincides both in plan and in section with tlie rest of 

 the road, and therefore conducts us in the easiest manner possible 

 from the one bank to the other. The skewed arch is inseparable from 

 the railway, as by its introduction alone the engineer is able to free 

 the line from awkward and injurious turnings. 



Having been consulted concerning the construction of an oblique 

 bridge of considerable magnitude, and never having met with anv 

 regular investigation into the theory of such structures, I was induced 

 to undertake the analysis. The results of that analysis 1 proceed to 

 hiy before the Society of Arts, in the hope that, though I m;iy be 

 wrong in supijosing them new, their publication may serve to dissemi- 

 nate correct notions on this intricate subject. It is a common idea 

 that the oblique is weaker than the right arch, and that the twist of 

 the stones causes a great waste of material. The truth is, that if 

 both bridges be skilfully constructed, there is no difl'erence in point of 

 strength between them, while the twist on the arch-stone of th(? 

 oblique bridge causes a most trifling loss of matter, :ui(l therefore our 

 road trustees should never liesitate to adojit that which agrees best 

 with the rest of the line. There is no limit to the obliquity, nor need 

 even the several abutnu'uts run parallel with each other. 



The general tpiestion of the construction of an arch resolves itself 

 into two parts; the first relating to the connexion which ought to exist 

 between the curvature of the vault and the weight piled on each por- 

 tion of it, is absolutely identical in the two cases of right and oblique 

 bridges, and is therefore left out in the present inquiry; the second, 

 however, relating to the forms of the arch-stones, bears directly on 

 the oblique arch, and will therefore engross almost our whole attention. 

 The outline of the bridge :nid the form of the vault Ir.iving been de- 

 termined on, the problem becomes tliis: To Co rer the surface of the 

 ceittcring with blocks of such sizts and forms as may insure the slabilili/ 

 of tlic structure. Now, if it be premised that the curved surface of 

 the vault must never bo vertical, the solution of the problem can 

 always be attained. 



It is clear, from the general form of a bridge, that the lines of pres- 



■' Ki'ad liclorc- llie Soiii-ly lor llic Kncourancincnt of the Useful Arls in 

 Si-ollamJ, un ISili NnuinUr and 2n(l Dcicniliur, liJSO ; 27lli January, 1H3G, 

 anil lUth May, l»3b. 



sure ought to run from one abutment to the other, and should be con- 

 tained in vertical planes parallel to the walls of the parapet. 

 Imagine, then, that the vault is intersected by a multitude of such 

 planes, the lines of intersection will indicate the directions in wduch 

 the pressures ought to be transmitted from block to block. Now the 

 stability of a structure is obtained by making the surfaces at which 

 the pressures are communicated perpendicular to the directions of 

 those jjressures, and therefore all that is required is to trace on the 

 surface of the centering a line which may cross all the lines of pres- 

 sure at right angles. In the case of the right arch, that line is a pa- 

 rallel to the abutment ; but in the oblique arch it becomes bent in a 

 pecidiar manner. 



At tlie crown of the cylindrical oblique arch, the joint-line is per- 

 pendicular to the parapet; of course, it begins to descend on the sur- 

 face of the vault, and as it descends it grailuallv bends away from that 

 direction to become more and more nearly parallel to the abutment. 

 If tlie crown line be regarded as the absciss, and the line of pressure 

 as the corresponding ordinate of the joint, the dilferential co-ethcient 

 of the line of pressure is in all cases proportional to the cosine of the 

 inclination which its extremity has to the horizon. If there be, then, 

 two closely contiguous joints, the portions of the lines of pressure 

 intercepted between them will be proportional to the cosines of the 

 obliquities, and hence it follows that the breadth (measured on a line 

 of pressure) of the stones in a given course dinunish in the ratio just 

 mentioned. It is a well known principle, that the strain upon any 

 arch stone is proportional to the secant of the same obliquity; and 

 thus, if the deptli of the stones be augmented to meet this increased 

 strain, it would follow that each voussoir in any given course ought to 

 exhibit the same extent of section by a plane parallel to the parapet. 

 The arch stones, both for convenience of workmanship and for appear- 

 ance, must be uniformly disposed from side to side ; and hence 

 throughout the whole structure they ought to be of uniform volume, 

 with the exception of the half stones left at the end of each alternate 

 course for the purpose of breaking the joint. The deepening of the 

 arch-stones toward the spring of the :irch is often, though very im- 

 properly, omitted ; in such c;ise the above statement does not hold 

 true. 



Even althouga the arch-stones were all equally broad upon the cen- 

 tering, those nearer the abutments would appear narrower on the 

 Gkound Plan, the breadths of their projections being proportional to 

 the cosines of their obliquity : hence the ground plan of an oblique 

 arch must present a very rapid diminution of breadths toward the 

 spring of the arch, the breadths of the projectious being, indeed, pro- 

 portional to the squares of the cosines of the obliquities. 



The Side Elevation of a vault with uniform voussoirs would ex- 

 hibit narrower intervals toward the crown, the breadths being propor- 

 tional to the sines of the obliquities; hence the side elevation of a 

 skewed arch must present narrow intervals both at the crown and at 

 the abutment, ;uid wider intervals upon the shoulders. The breadths 

 are proportional to the products of the sines by the cosines of the obli- 

 quities; that is, to the sines of twice the obliquities; and thus the 

 side elevations of those arch-stones which are inclined at 45" will be 

 the broadest. 



The End Elevation, or the projection of a joint upon the plane of 

 the parapet, possesses the very singular property of being entirely in- 

 dependent of the angle of the skew, and of depending alone on the 

 form of the longitudinal section of the ^'ault. This curious fact can 

 very readily be demonstrated. The projection of a right angle upon 

 a (ilane |iarallel to one of its sides is alw:iys a right angle, and there- 

 fore the projection of the joint u))on the plane of the parapet must 

 cross the projection of every line of pressure upon the same plane 

 perpendicularly. But the projections of all the lines of pressure are 

 equal to, and placed side by side with, each other, and are so what- 

 ever may be the angle of the skew, so that the delineation of the end 

 elevation of a joint, which requires only the tracing of a line that may 

 cross all these at right angles, will be performed exactly in the same 

 manner whether the bjidge be nmre or less oblique. When the angle 

 of obliquity diminishes to zero, that is, when the bridge becomes right, 

 the enil projections of the joints contract into mere points, wdiich 

 points are the commencements, so to speak, of the permanent curves 

 above mentioned. 



The end elevations of the beds of the voussoirs, or rather of the 

 lines formed by the intersection of these beds with the planes contain- 

 ing the lines of pressure, are also normals to the lines of pressure, and 

 must therefore be tangents to the end projections of the joints. From 

 tills it follows, that a short portion of a course, or a single arch-stone, 

 is very nearly contained between two planes slightly inclined to each 

 other; and that, therefore, the loss of material arising from the twist 

 o/ Me s/oKtinust always be iusignilicant. Those engineers who hav(! 

 experienced a loss on this account, hsvf done so because their bridges 



