1S40.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



197 



ON THE CONSTRUCTION OF OBLIQUE ARCHES. 



Sir — In your iiuinber for April, p. ll(i, I observe some observations 

 upon my work on Oblique Bridges, made by ;in anonymous writer 

 under the sicruLUure B. H. B., to whicli I feel disposed to make a reply, 

 requestincr t~lie favour of a place for it in your valuable periodical. 



In the first place I wish to premise that I tliink no author is under 

 the necessity of replying to the criticisms of an anonymo\is writer, and 

 that it would be more courteous if the writer of a paper professing to 

 be of a scientific character were to put it forth with his name. 



B. H. B. in alluding to myself says as follows : " he observes the lines 

 of the courses of the intrados should be made perpendicular to a line 

 drawn between the extremities of the face of the arch, without ever 

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

 than that it should be so." 



It is f|uite true that I did not assign a reason for this construction ; 

 because it is obviously in order that all the courses may be as nearly 

 as possible at right angles to both faces of the arch, and at the same 

 time parallel to each other. The greatest variation from the rectan- 

 gular intersection is at the middle of the development, or at the crown 

 of the arch: and at this point wdiere the course is nearly horizontal 

 the variation is of no practical iraportanct or objection: and it may 

 be shewn that it differs from a right angle by an angle whose tangent r= 



(-:-) 



cot e. 



The two methods suggested, proposed, or recommended by B. H. B. 

 to be substituted for the above, are most extraordinary. His second 

 method which he prefers, may be described as a recommendation to 

 build an oblique bridge by commencing ivith horizontal courses, and 

 "summering" them (in workman's pln-aseology) as the work rises 

 upon the centre. In this way the unscientific ugly old canal bridges 

 were built half a century back. B. H. B. concludes his short dis- 

 sertation on his proposed improvement in the following words: 



" The advantages to be derived from this are, first, that this angle 

 being less than that commonly employed, there will be less tendency 

 to slip ; and secondly, that being more nearly perpendicular to the face 

 of the arch, there is consequently more stability." 



Every thing herein contained is merely assumed ; and most cer- 

 tainly I venture to say that the stability of the oblique arch does not 

 depend upon the courses being laid at right angles to the line hounding 

 //le development : it is scarcely possible to conceive anything more 

 rotten than such a construction would prove. 



B. H. B. next says, "I am astonished at the serious errors into 

 which Mr. Buck has fallen in his last chapter, wliich is devoted to 

 further inrestigation, but which had better have been omitted alto- 

 gether. In attempting to determine at what altitute above the level 

 of the axis of the cylinder the thrust of the arch will be perpendicular 

 to the bed of the voussoir, he gives a formula which jiroduces the 

 strange result that the smaller the arch-stone, the lower will be the said 

 altitude, that is to say, the more secure will be the arch, and also that 

 it will be able to be built at a more acute 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 supported by 

 iron dowels and bolts : thus an arch built at an angle of 25° will require 

 no assistance from dowels, an arch built at 55° will require to be se- 

 cured by dowels to a height of 25" above the springing." I will carry 

 the quotation no further, because I shall now proceed to show that 

 "these errors" are attributable to B. H. B. and notto the formula. 

 For the information of those who have not read the work referred to, 

 I will here supply the general formula which I gave for the value of 

 sin T. as follows : 



Vr , r -\- e -. , / a . \ 2 1 a 



1(1— ^cos'«)-f (^^-^sm^e^ )-^^sin^e 



In this expression fl is the angle of obliquity, r is the radius of the 

 cylinder, c is the thickness of the arch, and t is the angle of elevation 

 of the point sought above the axis of the cylinder. 



Now if B. H. B. will look attentively at this expression, he will 

 see that its meaning is precisely the reverse of that which he has 

 stated ; for instance, " the smaller the arch stone " (or e is taken) the 

 greater will be the value of sin t ; and this is because e appears only 

 in the negative part of tlie expression. Again, the greater the value 

 of e or the thickness of the arch stone, the greater will be the negative 

 part of the expression, and consequently the smaller the value of sin t : 

 and the lower the point sought at which the thrust of the arch is 

 parallel to'its face. And consistently with this, " the greater the skew 

 of the bridge," the greater is the value of cos- 6 which is also found 

 only in the negative part of the expression, and consequently the 



Sin T — 



smaller will be the value of sin t, and " the less of the arch will have 

 to be supported by iron dowels and bolts." This result of the formula 

 is said to be a "strange phenomenon." I have no doubt it is very 

 startling to the reader, as I know it to be to every practical man at 

 first sight, but it is nevertheless true as I have satisfactorily determined 

 experimentally. I have constructed a model of a portion of an arch 

 at an angle of 25°, which is semicircular on the direct section: this 

 arch stands and keeps its form well without dowels, (although it is but 

 a narrow stripj, wdiereas one made to the same scale at an angle of 

 45° will not stand at all. 



B. H. B. proceeds to say, " the whole of these errors arise from 



having given the expression 



cosec fl cos T 



(nearly at the bottom of 



o-\ -ill- cot 9 cos T 

 page 3/) instead of ;- cosine (8 + (p) where <p is such an 



S IT 



angle that its tangent = 



cot 8 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, wdiich Mr. Buck has not taken into consideration." 



Here I most readily admit that I had omitted to take into consider- 

 ation the variable angle at which the courses intersect the face of the 

 arch. I discovered tliis defect about two months after the publication 

 of the work, and immediately prepared a correction for it, which is as 

 follows. I retain my former notation and the expression for the altitude 

 of the point C from which B. H. B. says the error arises, namely 



cosec e cos T 



1 , but in this case I shall substitute its equivalent for the 



segmental formula, or, ~ cosec 6 cos t, because the equation thence 



derived is general. I sliall now refer to the annexed diagrams : those 

 numbered 28 and 29 are identical with those to be found in my work ; 

 that numbered 27 is somewhat difierent. 



Fig. 29. 



Fiir. 28. 



i 7i, A.E 



Fig. 27. 



Let A B C in the annexed diagram, called fig. 27, represent the plan 

 of the acute quoin of the arch, then when the point A may have 

 ascended to the altitude signified by t, let us suppose it to be perpen- 

 dicularly over the point E, fig. 27. Let us also suppose A C tig. 27 



c ' 



which is sec 8, and CC fig. 28, which z=: - cosec 6 to remain constant, 



then C w hich is the summit of the tangent CC', fig. 28, will not be per- 

 pendicularly over the point/, fig. 27, (the extremity of E/ drawn 

 parallel to A C), but it will be at ^; here fg is the projection in plan 

 of the tangent C C ; now draw g A perpendicular to the face of the arch 

 B A, and to fulfil the conditions g h must be a horizontal line, and the 

 distance E h, considered as radius, if multiplied by the tangent of 

 IE K, tig. 29, must be equal to the altitude of ^ above E, tig. 27, or to 



Q 



- cosec e cos T. It now becomes necessary to determine an expression 



for the distance E h, and first E i is equal to A B by construction, there- 

 fore E A = Ei — ih, or (1 — ih.) 



