ARCHES.J BRIDGES 311 maximum rise and minimum thrust which can be drawn within the given middle third with the given loads. The line BC determines the point C, and the ordinates of BC give the ordinates of the curve DC, the ordinates being measured on corresponding verticals. The example shown is the diagram for an arch of 52 feet span and 10 feet 4 inches rise, the depth of the ring at the crown being 2 feet 6 inches, and at the springing 3 feet 8 inches. The loads per foot of breadth, beginning at a, arc 23 04, 19 45, IG 35, 13 94, 12-01, 10-35, 8-98, 7 93, 6 32, 5 79, 5 52, 5 39 cwts The rise of the .inear arch found is 10 2 feet, and h per foot of breadth of the arch 129 cwt. = - 1315 ft. cwt. 10-2 ft. 43. Empirical Expression for the Thickness of the Ring. The ring when not of equal thickness is always made of least depth at the crown. The depth of the key stone is therefore the thickness of the ring at its smallest part. Let D be this depth in feet, and r the radius of the arch in feet at the crown. Then we may take (Trautwine) 1 D = C r r . According to Rankine, C may be taken as 346 for a single arch, and 413 for one of a series of arches. The reason for making one of a series thicker than a single arch is, that the former has, when not loaded itself, to bear part of the thrust from its neighbours when these are loaded ; this thrust tends to throw the linear arch in the unloaded span low down in the keystone. The following is another series of values of C in practical use : For first class stonework C = 36 , , second class stonework C = 4 ,, brick and rabble C 45 Ferronet gives the following rule : Let L be the span in feet 2 D-nA Rankine, Civil Engineering, p. 427, shows that Trautwine s rule is rational. Perronet s can only be so when the usual proportion of rise to span is adopted. Brickwork arches of 24 feet span and less are made 1 foot 6 inches deep at the crown ; 30 feet span, 1 foot 104 inches ; 40 feet span, 2 feet 3 inches. The usual fiat arch of these dimensions has its ring increased by two rings of bricks towards the haunches. These do not show on the face being concealed by the spandrils. Rubble arches are made a little thicker. 44. Practical Details. The strongest and simplest form of arch is a flat circular arc, having a rise of about one quarter of the span. In these arches the springing is above the place where the joint of rupture would occur if the ring were prolonged. Those parts of an elliptical or semcircular arch which lie below the joint C, fig. 49, are of use chiefly to improve the appearance of the arch. They are virtually part of the abutment, which is some times even considered as extending to the joint B. In a very flat arch the linear arch may be brought to coincide more truly with the axis of the ring by lightening the haunch, with which object the roadway is sometimes carried on small flat arches turned at right angles to the main arch, and having the spandrils of the main arch as abutments. The joints between the voussoirs should be very evenly worked, so that the pressure may be evenly distributed. In brick joints the layers of mortar should be thin. Great care should be taken to provide for the drainage of the roadway above the arch. With this object the masonry should be covered with a sheet of asphalt sloping down to the piers or abutments, and suitable drains must be pro vided to collect the water and discharge it through the pier or abutment. SKEW ARCHES have already been treated of under the general head ARCH (vol. ii. p. 330). Considerable attention must be given to the con struction of the centres or wooden frames on which the voussoirs rest while the ring is in process of being built. Extreme rigidity is necessary, and this rigidity is best attained by adopting one of the three following plans Raukine): 1. Direct supports as in fig. 53, illustrating Hartley s centre for the bridge over the Dee at Chester (total span 200 feet) ; 2. Inclined struts in pairs ad shown in fig. 54, being a dia gram of the centre used fy -MJJ/ in the erection of Water loo Bridge ; 3. Trussed wooden girders, of which an example is afforded by the truss used in the erection of London Bridge, fig. 55. Figure 55 shows the striking plates and wedges by which the centre is lowered after the completion of the arch. The upper and lower plates A and B are strong Fig. 53. Fig. 54. beams suitably notched, and are separated by the com pound wedge C ; this wedge is kept in its place by cross wedges shown in section in the figure. When the centre Fi is to be lowered, these cross wedges are knocked out, and the main wedge C driven back. Owing to defective cen tering some large French arches sank much during con struction, and owing partly to this cause, and partly, as it would appear, to defective mason-work, the total deforma tion after the centres had been struck was most extraor dinary. In Perronet s bridge at Neuilly (vide Table XVII., 84) the sinking, while the centre was in its place, amounted to 13 inches, and after the centre was struck a further sinking took place of 9| inches. The crown of the centering had a radius of 150 feet, but the sinking of the arch was such that for GO feet it assumed the form of an arc of a circle with a radius of 244 feet. It is remark able that the bridge, built in 1774, of very bold design and so imperfectly executed, still stands. When the centres of Waterloo Bridge were removed no arch sank more than l inches. Centres have occasionally been sup ported on strong sacks full of sand. To lower the centre the sand was allowed to escape through apertures in the sack. It is believed that this method was first employed by a French engineer, M. Beaudemoulin. The canvas
sack has been advantageously replaced by wrought iron