Fiyure 2. LetABCDEFGbe the curve of an arch hich requires a Centre, ; first, form the equilateral polygon. A 11 c 0 E F G. with the timbers A B, a C. C D, &e..1111d fix the timbers A C, C E, E G. as belipre, which will form three triangles movable round A, C. E. G; let the timbers tI 1.) and n F fastened, and thus the whole will be immutable ; so that if supported at the points A and o, and it three applied at. any other of the angles 13, c, n, Or F, the timbers will all in a state of tension, or in a state of compression, and the whole may be looked upon as a solid body ; for suppose the tt jangle A n e to lie supported at the points A and n, the point c, and the other two sides, n c, c A. will be fixed ; and because n c is a triangle, and the points a and c ;ire fixed, the point 0, and consequently the soles C n and n a ; in like manner. sine,. c D E is a triangle. and the points c and D fixed, the point E will also be fixed, and therefore the sides D E and c. The same may be shown, in like manner, for the points F and o. Suppose, then, two equal and opposite forces applied at the points A and G, in the plane of the figure, the figure emit neither be extended out, nor compressed together. The pieces A 0, n n, and o t F, are alto other use than to make the centre stand firmly on its base. This disposition of the timbers will cause them to occupy the least possible space.
If the timbers are fixed at the points, A', I, o, p, Figure 2, the same immutability of figure may be demon strated ; for, suppose the points A and 11 to be fixed. the point A. will also be fixed ; the points A and k being fixed, the point a of the triangle A 13 ; again, the points n and k being fixed, the point 1 will also lie lixed : in the sglne manner, all the remaining points, o, F, p, G, I, will he proved to be in respect of the points A, II ; and the whole figure being kept in equilihrio by any three tierces. acting. in the plane of the figure, at any three angles. the action of the forces will only tend to compress or extend the timbers in a direction of their length.
In the construction of this truss, the triangular parts may be constructed all in the same plane, as in Fignre 1 ; and the a 1, and I) r may be halved upon the pieces c A and E G: but the utmost care must lie taken to secure the several pieces concurring at each if the angles, by bolting or iron straps, as no dependence can be put in any such joint without iron : but perhaps the best method of any is to halve the thickness of the pieces A C, C E, E G, at the 13111104 C and and also the pieces A 13, It c ; C D, n is; E F G ; at the points D. D, F: then bolting the ends A and c of the pieces 11 A, B C, the ends c and E if the pieces u e and n E. and the ends E
and o of the pieces F E and o. and than fixing double braces 13 D, D F, that is. fixing 13 D upon one side of the truss, and ma idler upon the other side of the truss, opposite to also fixing upon one side, and another opposite to it.
The disposition of the tiinbers fording only a series of quadrilaterals, gives nothing but immutability of figure. It can only derive its stiffness from the resistance of the joints.
Fiyure 3 shows the manner of forming a centre by two polygons, of *which the interior one is secured to the exterior: in this there is 110 occasion for double trussing-pieces, as the parts of the inscribed polygon act either as strutts or ties to that of the circumscribing one.
Figure 4 is the manner of tbrming the rib for a centre, by two independent trusses; in this ibrtn of centering there is no occasion tor bridles, or lonble trussing-pieces, as in those of Pitot, of the same construction.
Fiyure 5 is the manner of constructing a centre, according to Perronet, with four polygons, independent of each other, but with this improvement, that the lower extremities of each ring of polygons are framed into the two abutments ; this gives a much firmer base than if they were all to meet at the same place, and renders the centre-much stronger, by making the angles more acute. In this it becomes also necessary to have bridles, otherwise the exterior polygon only would be effective.
Figure 6 is the manner of constructing a centre with three polygons, which are all secured to each other. In this, truss pieces become necessary, otherwise the angles of the inner polygon would bend the sides of that next to it.
Plate II., Figure 1, is the design of a centre, its principle being that of two roofs intersecting each other. In this example, the forces which are communicated to the various parts of the frame are resisted longitudinally, either by corn pression or extension ; and no force is exerted transversely on any part, excepting the curved pieces in eontai•t with the boarding supporting tit arch-stones.
Figure :2 is the design of a centre ; it is first framed in one large truss, like a common roof, with two principal rafters. and a collar-beam ; each of the rafters becomes a tie for the two small trusses above, which arc framed in the manner of a roof', with queen-posts and braces. The lower angles of the principal rafters are braced from the lower queen-post: to the posts. This truss is free from transverse strains in a,l its parts, except the curve, which supports the arch-stones ; and, if well secured at the abutments, an arch of immense weight may be sustained by it.