18^7.] 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



211 



ON THE CONSTRUCTION OK ARCHES. 



A paper '■ Oit the existence (iiructically) of the line of equal Horizontal 

 Thrust in Arches, and the mode of determining it by Geometrical Construc- 

 tion." By William Henry Barlow, M. Inst. C.E. (Read al the 7«s/(- 

 tution ofCicit Lngineers. (With an Engraving, Plate XI ) 



The supposition of the existence of a certain curve or line, in which the 

 pressure is transmitted throughout the voussoirs of an arch, is not of recent 

 origin. The theory of equilibration, called the Catenarian, of which an ac- 

 count is given by David Gregory (Phil. Trans. 1697), is founded on this 

 basis ; but throughout the investigation, it has been assumed necessary to 

 make the line in which the pressure is transmitted, coincide precisely with 

 the form of the inlrados of the arch ; a condition which is necessary to sta- 

 bility, only when the arch is infinitely thin. 



In the theory promulgated by La Hire and Attwood, familiarly known 

 as the wedge theory, or that in which each voussoir is supposed to act as 

 a wedge, it is considered necessary that the pressure should be transmitted, 

 so that the direction in which it acts at each Joint, should be at right angles 

 to the surface of contact, which condition is only necessary to stability, 

 when no friction exists between the surfaces of contact of the voussoirs. 



But when the thickness of the arch and the friction at the surfaces of 

 contact of the voussoirs, are both included in the investigation, ii has been 

 shown by Professor Moseley, in his able and elegant exposition on this 

 subject, that the two conditions above mentioned, become modified, and 

 that in an arch of uncemented voussoirs, the actual requirements to estab- 

 lish stability are, — 



First, That the line in which the pressure is transmitted (which he has 

 named the line of resistance), should fall within the thickness of the arch 

 at every joint. 



Secondly, That the direction of the pressure, at each joint, should be 

 within certain limits, depending on the friction of the materials employed. 



Coulomb, the first writer on this subject, who based his assumptions on 

 data consistent with practice (Memoires des savans Strangers, 1773), con- 

 sidered, with Moseley, that there were two causes of rupture ; the first 

 arising from the turning over of certain parts of one voussoir on the edges 

 of another; and the second, from the slipping or sliding of the voussoirs 

 on each other; and although the mode of investigation pursued was totally 

 different, yet the results present a complete accordance with those since 

 arrived at by Professor Moseley, so far as they embrace the same elements 

 (discussion. This remark applies also to the catenarian and the wedge 

 theories; for if the thickness of the arch be considered to he infinitely sraall^ 

 the line of resistance becomes the catenary, and if the thickness be retained 

 and the friction omitted, the line of resistance is analogous with the line of 

 pressure as determined by Whewell in the wedge theory ; but though the 

 investigations of Moseley leave little to be done in elucidating the condi- 

 tions of stability in arches mathematically, yet the deductions have not re- 

 ceived that atteiiiion from engineers which iheir importance deserves 

 chiefly from the absence of any decided practical exhibition of their cor- 

 rectness and utility, and also from the investigation being surrounded by 

 too much mathematical difficulty, to admit of ready application. 



The analogy bcfiire-mentioned,a3 existing between the line of resistance' 

 the catenary, and the line of pressure of the wedge theory, arises from one 

 governing principle, which is general in these curves, and constitutes the 

 essential element of equilibrium when the only force acting is gravity 

 namely, that the horizontal forces in any part of the curve are equal to 

 each other ; by which it must be understood, that not only must the hori- 

 zontal force, at any part of the curve, be opposed by a horizontal force of 

 equal amount in the opposite direction, but that the horizontal force is equal 

 throughout the curve. This essential element of any curve of equilibrium 

 though probably known, has not been pointed out; its mathematical cor- 

 rectness is self-evident, and of its existence practically, as applied to the 

 line in which the pressure is transmitted through the voussoirs of an arch^ 

 the following experiments give satisfactory evidence:— 



In an arch composed of numerous voussoirs, let their surfaces of contact, 

 instead of being planes, be made curves, as in fig. 1. If the original form 

 of the arch be such that the line of resistance passes through the points of 

 contact, no motion will arise among the voussoirs, on removing the centre ; 

 but if the arch be a segment of a circle, or any other form which does not 

 coincide with the line of resistance, the voussoirs will take up a new posi- 

 tion, the curved surfaces of the voussoirs rolling on each other, to a certain 

 limit, when they come to rest, and if disturbed ,from this position (unless 



the disturbing force be sufficient to produce actual rupture), they will re- 

 turn to it.* 



Fig. 1. 



In this experiment It is obvious, that the pressure must be transmitted 

 through the points of contact ; and it atTords a practical proof, that this line 

 is the curve of equal horizontal thrust; for if in any voussoir a, the hori- 

 zontal force at i, was not equal to that at c, motion must ensue, and as this 

 condition is the same in all the voussoirs, it follows, thai the horizontal 

 force is equal throughout. The experiment admits of further application, 

 by loading the arch so as to vary the form of the curve in which the pres- 

 sure is transmitted, while it of necessity retains the element of equal hori- 

 zontal thrust; and it will be found, that the limit of stability is when the 

 point of contact of any two voussoirs falls at their outer or inner extremi- 

 ties ; thus establishing practically, that the line of resistance, or curve of 

 equal horizoulal thrust, must be contained within the thickness at every 

 joint. 



The second condition necessary to stability, namely, that the direction of 

 the pressure, at each joint, should be within the limiting angle of friction, 

 is almost always of necessity fulfilled in the forms of arches and with the 

 materials usually employed in practice; this part of the inquiry will there- 

 fore be confined to the first condition. 



Now the property of equal horizontal thrust, enables a geometrical con- 

 struction of the curve to be readily obtained in any given form of arch, if 

 two points in the curve be given, and by assuming these two points, it can 

 be ascertained by a tentative process, if any given arch does, or does not, 

 contain the curve. 



Proceeding in this manner it is found, that in a semicircular arch, the 

 thickness must be one ninth of the radius to contain the curve, a result 

 which is completely borne out in practice ; for though apparently unno- 

 ticed, a semicircular arch cannot be made to stand without foreign support, 

 unless the thickness be greater than one-ninth of the radius. 



In like manner, in any other form of arch which does not precisely coin- 

 cide with the curve of equal horizontal thrust, there is a certain minimum 

 thickness, or depth of voussoir, necessary to obtain stability. 



Among various other experiments, made to test the accuracy of the 

 theory, it will be sufHcient to give the following. The curve of equal hori- 

 zontal thrust, when drawn on the elevation of a semicircular arch, of which 

 the thickness is one-ninth of the ladius, touches the intrados at 35° above 

 the springing, and the extrados at the crown ; and practically, an arch of 

 these dimensions yields, by the crown descending, and the haunches going 

 outwards, the points of rupture, or rotation, being precisely those where 

 the curve touches the intrados and extrados. 



Fig. 3. 



* It is necessary that the radius of curvature of ttie voussoirs be made withio certain 

 limits, depending on the depth o( the voussoirs and the radius of the arch ; if too much 

 curvature be given, the arch will fall, before the points of contact can take up such a 

 position, as to coincide with the line of resistance. 



29* 



