1S40.J 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



109 



OF THE OBLIQUE OR SKEWED ARCH. 



While the system of communication from one part of tliR country 

 to anotliPr continued to be transmitted througli tlie medium of turupilce 

 roads alone, the instances were few and far between in wliicli tl'.e 

 erection of an uhlicjiie nr skt wed arch became necessary. Indeed, \ui- 

 less in very confined and precipitous situations, we do not recollect a 

 single case, where a structure of this kind has been resorted to for the 

 purpose of carrying a road over a river or slreauilet; nor was it re- 

 quisite that it should, for in laying down the original jilan of a road, 

 the surveyor would generally possess the power of directing it, so as 

 to intersect a river at right angles to its banks, and thus the necessity 

 of carrying a bridge obliquely across the stream would be altogether 

 avoided. 



On the introduction of canals however, the circumstances were very 

 materially altered, for it seldom happened that the direction of a road 

 already constructed, was permitted to be changed for the purpose of 

 acroiuiuodating it to the line of a projected canal, so as to traverse it 

 jierpendicularly ; and in many cases it would be found inconvenient if 

 not totally impracticable, to guide the canal across a road at right angles 

 to its diri'ction ; hence tlie necessity of having recourse to the skLipcd 

 arch, and accordingly, on the various canals that inteisect the country, 

 erections of this sort are very numerous, and the methods by which 

 some of tliem have been constructed are exceedingly ingenious. 



But it is in tlie construction of railroads that the skewed arch meets 

 with its Tuost important application, for in almost everv instance where 

 one line is intersected by another, the intersection takes place with a 

 lesser or greater degree of obliquity, and several viaducts of consider- 

 able length are wholly supported by a connected range of oblique 

 arcuation. This being the case, it is an object of the greatest im- 

 portance that the correct principles of construction should be rightiv 

 understood, and it is for the purpose of establishing thosi? principles 

 and rendering their application easy, that the ]u'esent investigation lias 

 been instituted. 



There are few architectural subjects that have excited a higher de- 

 gree of interest than the present, and there is none that has given rise 

 to a greater number of curious, abstruse and elegant theories, or been 

 the cause of more violent and protracted controversies. One party 

 contending that the just principle of construction, is to place the seve- 

 ral courses of which the arch is composed in a direction parallel to the 

 abutments, the direction of the coursing joints being regulated by the 

 nature of the curve on which the arch is built. A second party main- 

 tains, that the several courses should be placed perpendicular to the 

 face of the arch as far as the obliquity on both sides of it, and that the 

 middle portion which stands upon the square, should have the courses 

 laid parallel to the imposts or abutments. A third class of disputants 

 insists upon laying the several courses perpendicular to the face of the 

 arch throughout its whole extent, and trending them to the abutments 

 in an angle dependent on the given obliquity; while a fourth class 

 proposes to direct the courses in such a manner as to traverse the arch 

 spirally like the threads of screw. 



The subject itself is worthy of a mechanical investigation, and since 

 we have been induced to direct our ittention to it, we shall endeavour 

 to the utmost of our power to set the question at rest, and point out 

 the true principles of construction upon which depends the maximum 

 of stability and strength. 



In taking a minute and comprehensive view of the subject to which 

 our present enquiries are directed, it will be proper for the sake of 

 system, to consider the various theories above specified in the same 

 order as we have described them. This in the first place will lead us 

 to the contemplation of that variety where the courses are laid in a 

 direction parallel to the imposts, and in which, (supposing the arch to 

 be a semicircle,) the planes of the coursing joints on being produced 

 to intersect the plan or base of the arch, are everywhere constrained 

 to terminate in the axis or straight line, which passing through the 

 centre of the semicircle divides the plan into two equal and similar 

 portions. 



The princi|)le upon which the mechanical delineation of this parti- 

 cular form is founded, is exceedingly curious and interesting, referring 

 as it does to the developement of the several parts of a right angled 

 triangular pyramid upon a plane surface. This circumstance intro- 

 duces a species of calculation that is not generally understood by prac- 

 tical architects, since it claims as its basis the doctrine of Spherical 

 Trigonometry, a subject to which the attention of practical men is very 

 seldom directed, although its applications are both numerous and im- 

 portant, and its principles remarkable for their elegance and simplicity. 

 The objects of calculation are, the angles at the vertex of the pyramid 

 comprehended between its edges, and the angles which measure the 

 mutual inclinations of its bounding planes. Now, in order to assimilate 



the necessary operations to the determination of the levels or moulds 

 by which the several voussoirs or arch stones are framed, we have 

 only to consider the nature of the figure arising from the nmtual inter- 

 sections of the planes to which the moulds are severally applied. 



(f the i'ace or elevation of the arch, and the planes of the coursing 

 joints or beds of the several voussoirs, be produced to intersect each 

 other in the plan or base on which the arch is raised, they will, in con- 

 nexion with the said plan, manifestly constitute a series of triangular 

 jjyramids having their vertices in the centre of the semicircle, and if 

 the face of the arch be perpendicular to the plan, the pyramids will 

 be all right angled ; that is, two of the boiniding planes in each, namely, 

 the face and plan of the arch will intersect one another in an angle of 

 ninety degrees. 



Let the planes of the beds or 

 coursing joints be produced ex- 

 ternally, and conceive a circular 

 arc to be described in each of 

 the bounding planes, and having 

 the vertex of the pvramid ;is a 

 centre ; then, the figures thus 

 constituted will respectively re- 

 semble that which is exhibited 

 in the margin, and upon the de- 

 velopeauMit of wddch the con- 

 struction of the arch depends. 



A, part of the arch. P, part 

 of the plan. B, part of bed pro- 

 longed. 



If the middle plane or plan 

 s C D be supposed to be fixed, 

 w hile the extreme planes ;• C s 

 and D C I are elevated about the 



lines C s, C D, till the points )• and >, as also the radii C )• and C / 

 coincide, the nature of the figure thus formed will become manifest, 

 and th(< expansions of its several parts \ipon a plane surface, may 

 be etlected in the fcjUowing manner. 



With the chord of GO de- -r 



grees taken from a scale of 

 any convenient magnitude at 

 pleasure, and about C as a 

 centre, describe the circular 

 arc rsD/, upon which and 

 from the same scale of chords, 

 set off ;• s and s L), respectively 

 equal to the measures of the 

 angles at the vertices of the 

 perpendicular planes cC s and 

 sCD. 



Driiw the radii C ;■, C s and 

 C D, and in the radius C r 

 take any point a at pleasure, 

 and erect the perpendicular 

 a A meeting the radius Cs in the point A. At the point A deter- 

 ndned in this manner, erect the perpendicular A D meeting the radius 

 CD in the point D. From A, and on the radius Cs set oft' A/ equal 

 to A a and draw Dy. Upon CD as a diameter, describe the semi- 

 circular C H D, in which lay oft' the chord C^ equal to C a, and D g 

 equal to Dy' and draw the radius Cl. 



The above operation developes the triangular pyramid as far as it 

 relates to the construction of the arch in question ; D C I being the 

 bevel of the bed or coursing joint, and Ay D the bevel between the 

 coursing joint and face of the arch. But in order to exhibit the com- 

 plete developement of the figure, it is necessary to determine the angle 

 which measures the inclination of the planes s C D and DC/; that is, 

 the angle contained between the plan of the arch and the bed of the 

 voussoirs for any particular course. From A or any other point wdiat- 

 ever in the radius C s, let fall the perpendicular A b, carrying it for- 

 ward to meet C / in d; then is A 6 the base, and bd the hypothenuse 

 of a right angled plane triangle, between which the required angle lies. 

 At the point A in the straight line d A, erect tfie perpendicular A c, 

 and make be equal to hd; then is A 6 c the angle sought, which hav- 

 ing been found, the developement of the pyramid is complete. 



The nature and principles of the above construction will be readily 

 perceived by reversing the process; that is, by recomposing the figure 

 from its constituent planes and the angles which measure their in- 

 clinations: and for this purpose, let the two extreme planes rCs and 

 D C / be turned about the radii C s and C D, while the ndddle plane 

 s C D remains fixed ; and at the same, let the triangular planes A/D 



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