1847.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



213 



The first of ihese couslructions, is that which is more particularly appli- 

 cable, in determiniHg whether a given form of arch contains the curve ; for 

 by taking each joint separately, the whole curve is obtained. The second 

 is that which is employed, in determining whether a given abutment is of 

 sufficient thickness to contain the curve. 



Problem I.— Let a and b, fig. 0, be two points in the curve of equal 

 horizontal thrust in the arch A B ; required to find the point at which the 

 curve intersects the joint o q. Let G be the centre of gravity of the half 

 arch A B, and g- that of the portion o 9 B. Through b draw the horizontal 

 line c n, and the vertical line b I ; also through G and g, draw the vertical 

 lines G h and g k. intersecting en in h and k ; join a h, and produce it to 

 J; from k set off fc )i, equal to A 6, and through h draw the vertical line 

 n'm, making 11 miolb as the weight of the portion o 9 B, is to the weight 

 of the half arch A B ; join m fc and produce it until it intersects oq; p, the 

 point of intersection, will be the point required. 



I 



Rg. 7. 

 Problem II. — Let a 6, fig. 7, be two points in the curve of equal hori- 

 zontal thrust, in the arch A B ; required to find the point at which the curve 

 intersects the joint q, being the base of the abutment. Let G be the centre 

 of gravity for the half arch A B, and g- that of the arch and abutment taken 

 together. Through h, draw the horizontal line 6 r, and the vertical line 

 b I ; also through G and g, draw the vertical lines G h and g- k, intersecting 

 br ia h and k ; join a h and produce it to I, from k set offfcn equal to ft b, 

 and through n draw the vertical line n m, making w »n to i 6 as the weight 

 of the arch and abutment is to the weight of the arch A B ; join m k, and 

 produce it, until it intersects « ^ ; p, the point of intersection, will be the 

 point required. 



It is unnecessary to accompany these constructions with a demonstration, 

 as it is evident, from the nature of the construction in either case, that the 

 horizontal thrust of the portion A B, at the points a and b, is equal to that 

 of 7 B, at the points p and 6. — For a loaded arch the construction remains 

 the same ; the centre of gravity of the arch and load being taken, instead 

 of that of the arch only. 



These constructions point out, not only the form of the curve of equal 

 horizontal thrust in any given arch, but also the direction and amount of 

 pressure at any joint. For as the perpendiculars of the several triangles 

 represent the weights of the several parts, so the hypolhenuse of the seve- 

 ral triangles represent the resultant pressures at any joint. From this it 

 appears, that the actual pressure, tending to crush the material of which 

 the arch is made, decreases towards the crown of the arch. 



Figs. 8, 9, 10, 11, and 12 (Plate XI.), are drawings to scale, of ordinary 

 forms of arches, showing the minimum thickness that will contain the 

 curve of equal horizontal thrust, and that this is the least thickness capa- 

 ble of standing practically, may be readily tested by models ; due allow- 

 ance being made, on account of the joints not being able to be worked 



with mathematical exactness. From these diagrams it appears, that th „ 

 arches which differ most in form from their curves of equal horizontal 

 thrust, are semicircles and semi-ellipses, and that in these forms, there is a 

 tendency for the crown to descend, and the haunches to go outwards. 

 Hence the utility and the general adoption of solid backing and spandril 

 walls in these forms of arches. The pointed arch has a tendency to go up 

 in the crown. 



Figs. IS and 14 show the variations produced in the curve of horizontal 

 thrust, by the addition of the tilling in, up to the level of the roadway. 



Hitherto, only one line, or curve of equal horizontal thrust, has been 

 spoken of; but if the thickness of an arch be more than sufficient to con- 

 tain this curve, it is obvious, from the nature of the construction, that more 

 than one such curve will be contained in it, and if the theory advanced is 

 correct, the arch ought to be capable of being supported in any one of 

 these curves. 



Fig. 15. 



The truth of this position was practically tested by the model repre- 

 sented in fig. 15, which consisted of an arch composed of six voussoir=, 

 separated at each joint by four small pieces of wood, each of which could 

 be withdrawn by hand. A curve of equal horizontal thrust was theu care- 

 fully drawn upon the profile of the arch, as represented in the figure by 

 the line a ic, and it was found that, provided the separating pieces were 

 left in at the points where this curved line intersected the surfaces of the 

 voussoirs, the whole of the remaining pieces might be removed, without 

 producing rupture of the arch ; in the same manner it could be supported 

 in the curves df or g hi, or in any curve of equal horizontal thrust, 

 which was contained within the depth of the voussoirs ; but that if the 

 separating pieces were so placed, that a curve of equal horizontal thrust 

 could not pass through every one of them, the stability of the arch could 

 not be maintained. Of these curves there are two limits, namely, that in 

 which the ratio of the versed sine to the chord at the springing is the 

 greatest, and that in which it is the least. These two curves are repre- 

 sented in fig. 16 (Plate XI.), and are both determinable by the same pro- 

 cess. 



The first points out the curve, in which the pressure is transmitted 

 through the voussoirs to the abutment, and is identical with that called by 

 Moseley, " the line of resistance." 



The other points out the curve, in which a pressure from without would 

 be transmitted from the springing through the arch ; such as would arise 

 from the thrust of a second arch. This line may be called, for the sake of 

 distinction, " the line of impression." The one curve, in short, is derived 

 or generated by the pressure the arch exerts; the other that which it is 

 capable of resisting. In diflerent forms and constructions of arches, the 

 amounts of these forces vary very greatly, and it becomes a consideration 

 of importance, where arches of different sizes are abutted against each 

 other. 



In the flat arch, fig. 17, the line of impression is a straight line, and 



Fig. 17. 

 therefore, equilibrium could not be destroyed by outward horizontal pres- 

 sure, until the material yielded by crushing ; while by increasing the depth 

 of the voussoirs, the thrust exerted on the abutments may be diminished 

 and rendered comparatively small. 



From this, a knowledge of a property in arches^ is arrived ,'at, which 



