no 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[April, 



SKEW ARCHES. 



Sir, — I am surprised that among the many correspondents who ad- 

 dress yi)U, there are so few of them either theoretical or practical, who 

 touch iipoi) tlie siil)ject of slvew arches, a subject wliich ))re5en(s so 

 wide a field of observation and reniarlc. 



Among the very few works which we possess on this point, Afr. 

 Buck's seems t(i hold tliehitjhest place, although even in it there seems, 

 to me several things which would be the better for alteration or 

 amenilment. Altliough he is ])articular in giving the mathematical 

 formula fur calculating the necessary angles and lines, yet lie assumes 

 some things, as granted, which lie at the very foundation of his prin- 

 ciples; for example, he observes tliat the lines of the courses of (he 

 intrados should be made perpendicular to a line drawn between the 

 extremities of the developement of the face of the arch, without ever 

 giving any reason for it, or making any remark on the subject, farther 

 than that it sbould be so. Now it strikes me that a considerable alter- 

 ation maybe made in this for the better. LetAHCDEFCin the 

 f.gnre aiuiexed) be the developement of a semicircular arch, then 



there is a curve A G H, such that a tangent drawn from any point in 

 this curve is perpendicular to the face of the arch at the said point, 

 as, the tangent G K, drawn from the point G is perpendicular to the 

 developement of the face of the arch B, G C, at the said point G. Now 

 if the courses were drawn similarly to this as shown in that part of 

 the figure A, B, C, F, then the arch (according to Mr. Buck, in the be- 

 ginning of liis seventh chapter) would be perfectly secure. Unfor- 

 tunately however, the difhculty of execution would be so great, if it is 

 not an impossibility, that this could never be applied vigorously to 

 practice, and the only way left is to make the best practicable ap- 

 proximation to tills curve. There are two methods, either of which 

 appear to me to be better than tliat of Mr. Buck's, although the first 

 has a considerable drawback, because the beauty of the arch is very 

 much destroyed on account of the unequal divisions of the courses. 

 The first method is after having drawn a line as F C perpendicular to 

 the face of the arch at the centre, to divide the segments F E, and 

 C D into an equal convenient number of parts, and to draw the courses 

 as shown by dotted lines from the one face to tlie other through the 

 respective points 1, 1 — 2, 2 — 3, 3, &c. This, although a little more 

 expensive than tlie common method, appears to me more desiraljle on 

 account of the additional strength which it possesses. I may mention 

 that I was shown a model built upon this principle, which when sub- 

 jected to a pressure on the crown, forced the abutments asunder ex- 

 actly in the line of the face of the arch, thus giving the best proof of 

 the correctness of the princi|)le. The second method which I would 

 recommend is simply instead of drawing the line of the intradosal 

 courses perpendicular to the straight line A E, to draw it nearly 

 averaging the curve A G H, tlie tangent of the angle which such a 



cot e 

 line would form with the abutments approximates to —x—. e being 



the angle of the acute corner of the abutments. The advantages to 

 be derived from this are, first that this angle being less than that com- 

 monly employed, there will be less tendency to slip, and secondly, that 

 being more nearly perpendicular to the face of tlie arch, there is con- 

 sequently more stability. 



1 am astonished at the serious errors into which Mr. Buck lias fallen 

 in his last chapter, which is devoted to farthtr iuKsstii^atiom, but 

 which had better have been omitted altogether. In attempting to 

 determine at what attitude above the level of the axis of the cylinder 

 the thrust of the arch will be perpendicular to the bed of the voussoir, 

 be gives a formula which produces the strange result that the smaller 

 the archstoiie, the lower will be the said attitude, that is to say, the 



more secure will be the arch, and also that it will be able to be built 

 at a more acufe angle. Another still more strange phenomenon, the 

 result of this formula, is that the greater the skew of the bridge the 

 less of the arch will have to be supi)orted by iron dowels and bolts ; 

 thus an arch built at an angle of 25" will require no assistance from 

 dowels, an arch built at an angle of S.^'J will require to he secured bv 

 dowels to a height of 25" above the springing, and lastly, an arch cif 

 90" or square to the abutments, will have to be secured to a height of 

 40° above the springing. The w hole of these errors arise from having 



given the expression 



. ^ , r cot 9. cos T 

 instead of , 



cosec. e. cos T 



, (nearly at the bottom of page 37), 



-f- cosine (6 -|- ip) where ip is such an angle 



that its tangent is =: 



cot e. sin T. 



This must be evident to any one 



who considers that the courses alter their angle with regard to the 

 face of the arch, which Mr. Buck has not taken into consideration. 



As it may be of some use to settle this |irobleiu, I would submit the 

 following solution, observing that the letters and charaters refer to the 

 same as in Mr. Buck's treatise. 



1st. Ill finding a term for C O, I would reject the thickness of the 



cylinder, and consider the jioint C) as that to which the tangents of the 



small curves which show in the face of the arch, tend ; this i« more 



correct because the joints of the voussoirs being segments of curves, 



there can be no point on the face of the arch at wHiich a ball would 



roll down the bed into a line exactly jiarallel to the face; this may be 



considered too minute for observation, but besides being more correct 



it will simplify the cjuestion much. 



cot' 6 

 Then upon this ground C O = — ^— , and taking Mr. Buck's own 



cot' 6 -\- V. sin T 



fieures at the bottom of iiage 37. I E K = 

 ° V. cos T. cosec 9 



2d. In finding the tangent of the internal angle, Mr. Buck states 

 correctly " that the tangent of the angle, which the tangent of the in- 

 tradosal spiral makes with the horizon diminishes as cos. t," but he 

 has omitted to mention that the angle 9, which the course makes with 

 the face at the springing, increases as a certain function of sin. t be- 

 coming (9 -j- <p), where (p is such an angle that it has for a tangent 



— '—'- '— ; this then would make tlie tangent of the internal angle 



cot 9. cos T 



at the point sought -= ^^ . ^, 



^ " i IT. COS (9 -\-<p) 



external and internal angles, we have 



then equating these values if tlie 



cot 9. cos T 



cot' 9 



-]- f. sin T. 



g IT. cos (9 -(- (p) V, cos T. cosec 9 

 but rejecting f in the second side of the eq\iation, because by hypo- 

 thesis' it is unitv, and multiplying both sides by \ ir, cos. t, cosec 9, 



cot 9. cos" T ., a 1 1 



we have — : -rr-, = cot- 9 -\- it -k. sin t. 



sin 9. cos (9 + (p) 



After this the solution must be completed by a series of approxima- 

 tions luitil a true value of t can be found. If the thickness of the 

 archstones is wished to be considered, then by making the second side 

 p -\- e 



of this last equation cot- 9. 

 quired result, thus. 



-J- i TT. sin T. it will give the i e- 



T when C O 



cot- 9 , ^. ,^ <""'' ® 



, r, or T when C U = -t — • 



V +e. 



When 9 is 60 then 40 56 - - - 40 

 45 - 42 46 - - - 40 

 30 - 50 10 - - - 43 

 The numbers in the last column are only approximations, but it may 

 be taken that in all arches of a moderite skew, the point r is about 40° 

 ;ibove the level of llie axis of the cylinder. 



I have merely thrown out these observations for the purpose of 

 direcling attention to this particular kind of arch, which is now come 

 into such common use, and about which we have so little information, 

 and that little of a very loose kind with regard to the theory of the 

 arch, but I think that Mr. Buck is entitled to the thanks of the pro- 

 fession for the clearness and accuracy with which he has explained 

 anil illustrated the greater portion of the subject practically considered. 

 I remain. Sir, vour's respectfully, 



B. H. B. 



Edinburgli, March, 1840, 



