2S0 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[August 



voussoirs, where the points of contact, and consequently the curve of pres- 

 sure, varied according to the spot where the pressure was imposed. In 

 praclice, this pointed out the necessity of adapling the form of the arch to 

 the service it was intended to perform. For instance, if the roadway over 

 au arch were level, and the pressure equal, the fracture would take place 

 by depression of the haunches and the opening of the extrados of the 

 crown; but if the roadway were curved, the pressure being thrown upon 

 the crown, the crown would fall, and openings would occur at the extrados 

 of the bauuches. Different calculations must therefore be made for the 

 dilferent constructions. 



Mr. G. Snell observed, that Mr. Barlow, in his geometric conlruction of 

 the line of resistance, assumed, that there were already two points found. 

 Now two points in the line of resistance were determined, when the points 

 of rupture were known; for, at the time when rupture was about lo take 

 place, when the arch was balancing between standing and falling, the line 

 of resistance touched the extrados or intrados of the arch, at the points of 

 rupture. One of these points of rupture was determined by the conditions 

 of the question, the other might be determined by a geometric construction, 

 founded on the principles set forth in Moseley's works, and which he had 

 demonstrated in his lectures. The process was one of approximation, and 

 he supposed three trials would be sufficient to determine the point of rup- 

 ture, with pel feet accuracy. The process would apply to all shapes of 

 arches under pressure, in any direction, in any position, or of any amount ; 

 but Mr. Snell would at present confine himself to the simple case of an arch 

 which was loaded equally on either side, and the voussoirs of which were 

 equal each to each, on either sideof the centre line, lu such a case, the one 

 point of rupture would be at the crown of the arch, which would be on the 

 point of turning on one of its edges, at its extrados, if the arch was about 

 to fail by the sinking of the crown ; and at its intrados, if it was about to 

 fail by the rising of the crown. In the first case, some stone at the 



haunches would be on the point of 

 turning on its edge at the intrados. 

 In the second case, some stone at 

 the haunches would be on the point 

 of turning on its edge at the extra- 

 dos. He would confine himself to 

 the first of these cases. Then, to 

 find the point of rupture, choose 

 some point C (D?. 1), which was 

 considered to be near the point of 

 rupture, and which, in this case 

 was at the intrados ; draw C D the 

 joint of the voussoir. The arch 

 being about to fail, by the turning 

 of the key-stone on its edge at A, 

 the resultant of all the forces, at that 

 Fig. 1. point, must touch the curve of the 



extrados at A ; its direction was therefore horizontal, and was represented 

 in position and direction by the line Ae. The resultant of the weight of 

 the mass A, B, C, D, was represented in position and direction, by a verti- 

 cal line passing throuuh the centre of gravity, G, such as G It. Now, the 

 only forces acting on the point C (in this case) were the resultant of the 

 forces at A, and the weight of the mass A, B, C, D, and these, being re- 

 presented in position and direction respectively, by A e and G A, \vliich 

 intersected at the point m, the resultant of all the forces acted through m ; 

 it also acted at C, and therefore ;» C represented the resultant of all the 

 forces acting at C. Now, as before stated, if C was the point of rupture, 

 the line of resistance touched the curve of the intrados at C, therefore a 

 tangent to the line of resistance at C was aUo a tangent to the intrados, and 

 the^esultaot of all the forces, acting on any point in the line of resistance, 

 was in the direction of a tangent to that line. Therefore, if C was the 

 point of rupture, m C was a tangent to the line of resistance, and therefore 

 HI C was a tangent to the intrados, as in fig. 1. If G was nut the point of 

 rupture, but if the point of rupture was above 0, m C would cut the in- 

 trados, as in fig. 2 ; but if the point of rupture was below C, mC would 

 rut the intrados, as in fig. 3. 



Fig. 2. 



F!g. 3 



also adopted a tentative process. Moseley's line of resistance touched the 

 intrados and extrados at the points of rupture. Now a practical eye could 

 detect very closely which would be the point of rupture, and a curve of 

 equal horizontal thrust drawn through these points, though it might not 

 produce the line of resistance with mathematical accuracy, was sufliciently 

 near it for all useful purposes. Moseley's theory was undoubtedly very 

 perfect ; in fact he was the only mathematician wlio had treated the sub- 

 ject consistently with its practical requirements. The dilficulty in his 

 mode of investigation was in those arches which did not partake of regular 

 geometric forms, and in these cases Mr. Barlow's method would be found 

 easy of application. 



Mr. Brusel still thought, that Mr. Barlow had scarcely met the objec- 

 tions which had been raised. It was true that in practice some points 

 might be assumed ; but it was more satisfactory to have positive rules for 

 finding these points, and assuring the mind as to the correctness of the 

 basis of the proposition. In a very large arch, with a small rise, the line 

 of pressure must be confined within very narrow limits, and in such a 

 case a formula giving the points definitively was essential for inspiring 

 confidence. 



Mr. W. H. Barlow replied, that the limits %Thich confined the line of 

 resistance, depended on the thickness of the arch and not on the ratio of 

 the rise to the span ; the points of rupture in ordinary forms of arches were 

 well known; they were at the extrados in the crown, and at the intrados 

 in the haunches ; there was, therefore, no difficulty in finding the line of 

 resistance in these cases. If the mind was as perfectly impressed with the 

 direction of the forces in arches, as in the case of columns, both could be 

 built with equal security. 



Mr. G. Snell stated, that in all cases of equal thickness of voussoirs 

 throughout, Mr. Barlow's rules might apply; but if the thickness was less 

 al the crown, as in the case of au arch with a keystone of limited depth, 

 but of which the voussoirs increased towards the abutments until they 

 came to an extreme length, he did not see where Mr. Barlow could assume 

 his points in the line of resistance. 



Mr. W. H. Barlow replied, that in reference to that particular form of 

 arch, it was evident many curves of equal horizontal thrust would be 

 drawn within the thickness, so that it was unnecessary to entertain the 

 question ; because, if any one curve of equal horizontal thrust was contain- 

 ed, it proved that the theoretical line of resistance was also contained. It 

 would be observed, on referring to the paper and consulting the drawings 

 and models, that the rules were general, and applied to every form of arch 

 and arrhiform structure, loaded or unloaded, and whether of equal thick- 

 ness or otherwise. The model, with tlie rectangular voussoirs leaning to- 

 gether at the apex, was selected as an extreme case. He wished to re- 

 move an impression, which might have been produced, by his slating that 

 his mode of treating the subject of arches was not mathematical as that of 

 Professor Moseley : he only alluded to the use of geometric construction 

 instead of algebraic formula ; the principle or theory was the same in both 

 cases. The misapprehension as to assuming points in the curve, which 

 Mr. Stephenson alluded to, as not having been sufficiently explained, arose 

 from the modification which was necessary in applying theory to practice. 

 If perfect hardness of materials and mathematical accuracy of workman- 

 ship were attainable, the pressure would be transmitted in the line of re- 

 sistance, as laid down by Moseley, and described by Mr. Snell. On the 

 other hand, if the materials were in the softest slate in which it was possi- 

 ble for au arch to sustain itself, the pressure would be transmitted in that 

 curve of equal horizontal thrust, which corresponded most nearly to a line 

 drawn through the centre of the thickness of the arch ; because in that 

 state of the arch, the whole available surface of the voussoirs must be act- 

 ing, to support the insistent pressure ; practically, therefore, the pressure 

 would be transmitted in a curve of equal horizontal thrust, somewhere be- 

 tween these two limits. Now, in the case of large brick arches, particu- 

 larly when the centres were first struck, the state of the arch approached 

 that which had been just mentioned, and it was for that reason he had 

 stated in the paper, that in determining abutments for arches of large dimen- 

 sions, the points p ;>' should be taken in the centre of the thickness of the 

 arch, though the extreme limit of stability, if the materials were hard, 

 would be when the points p ;/ were in the theoretical line of resistance. 

 Assuming the points p ]>' to be in the centre of the thickness, and making 

 the abutments accordingly, was in ellect nothing more than providing abut- 

 ments of such dimensions as should resist the thrust of the arch, when it 

 was in the most disadvantageous state in which it was possible for it to 

 exist. An arch constructed with abutments only just sutlicieut to contain 

 the theoretical line of resistance, would possess the same degree of stability 

 as a column placed so far out of perpendicular, that a vertical line drawn 

 through its centre of gravity would just fall at the extremity of its base ; 

 but au arch, with abutments built so as to contain the curve of equal hori- 

 zontal thrust, which accorded nearest to the centre line of its depth or 

 thickness, would be under the same condition of stability as a column 

 placed perfectly vertical. 



Mr. H. W. Barlow said, he assumed two points in order to facilitate 

 the investigation. As far as practical utility was concerned, the line of 

 thrust might thus be obtained at one operation, instead of pursuing the la- 

 borious process necessary for determining the theoretic line of resistance ; 

 <ndeed,exceptingfortbe most regular geometric forms of the arch, Moseley 



