Fig. 107 is a design for a centre for a stone bridge where intermediate supports, as piles driven into the bed of the river, are practicable. Its timbers are so distributed as to sustain the weight of the arch-stones as they are being laid, without destroying the original form of the centre; and also to prevent its destruction or settlement, should any of the piles be swept away. The most usual error in badly-constructed centres is that the timbers are disposed so as to cause the framing to rise at the crown during the laying of the arch-stones up the sides. To remedy this evil, some have loaded the crown with heavy stones; but a centre properly constructed will need no such precaution.

247 Centres For Stone Bridges 138

Fig. 107.

Experiments have shown that an arch-stone does not press upon the centring until its bed is inclined to the horizon at an angle of from 30 to 45 degrees, according to the hardness of the stone, and whether it is laid in mortar or not. For general purposes, the point at which the pressure commences may be considered to be at that joint which forms an angle of 32 degrees with the horizon. At this point the pressure is inconsiderable, but gradually increases towards the crown. The following table gives the portion of the weight of the arch-stones that presses upon the framing at the various angles of inclination formed by the bed of the stone with the horizon. The pressure perpendicular to the curve is equal to the weight of the arch-stone multiplied by the decimal -

.0, when the angle of inclination is 32 degrees.

.04

,,

,,

,,

34

,,

.08

,,

,,

,,

36

,,

. 12

,,

,,

,,

38

,,

.17

,,

,,

,,

40

,,

.21

,,

,,

,,

42

,,

.25

,,

,,

,,

44

,,

.29

,,

,,

,,

46

,,

.33

,,

,,

,,

48

,,

.37

,,

,,

,,

50

,,

.4

,,

,,

,,

52

,,

.44

,,

,,

,,

54

,,

.48

,,

,,

,,

56

,,

.52

,,

,,

,,

58

,,

.54

,,

,,

,,

60

,,

From this it is seen that at the inclination of 44 degrees the pressure equals one quarter the weight of the stone; at 57 degrees, half the weight; and when a vertical line, as ab (Fig. 108), passing through the centre of gravity of the arch-stone, does not fall within its bed, c d, the pressure may be considered equal to the whole weight of the stone. This will be the case at about 60 degrees, when the depth of the stone is double its breadth. The direction of these pressures is considered in a line with the radius of the curve. The weight upon a centre being known, the pressure may be estimated and the timber calculated accordingly. But it must be remembered that the whole weight is never placed upon the framing at once - as seems to have been the idea had in view by the designers of some centres. In building the arch, it should be commenced at each buttress at the same time (as is generally the case), and each side should progress equally towards the crown. In designing the framing, the effect produced by each successive layer of stone should be considered. The pressure of the stones upon one side should, by the arrangement of the struts, be counterpoised by that of the stones upon the other side.

247 Centres For Stone Bridges 139

Fig. 108.

Over a river whose stream is rapid, or where it is necessary to preserve an uninterrupted passage for the purposes of navigation, the centre must be constructed without intermediate supports, and without a continued horizontal tie at the base; such a centre is shown at Fig. 109. In laying the stones from the base up to a and c, the pieces bd and bd act as ties to prevent any rising at b. After this, while the stones are being laid from a and from c to b, they act as struts; the piece f g is added for additional security. Upon this plan, with some variation to suit circumstances, centres may be constructed for any span usual in stone-bridge building.

247 Centres For Stone Bridges 140

Fig. 109.

In bridge centres, the principal timbers should abut, and not be intercepted by a suspension or radial piece between. These should be in halves, notched on each side and bolted. The timbers should intersect as little as possible, for the more joints the greater is the settling; and halving them together is a bad practice, as it destroys nearly one half the strength of the timber. Ties should be introduced across, especially where many timbers meet; and as the centre is to serve but a temporary purpose, the whole should be designed with a view to employ the timber afterwards for other uses. For this reason, all unnecessary cutting should be avoided.

Centres should be sufficiently strong to preserve a staunch and steady form during the whole process of building; for any shaking or trembling will have a tendency to prevent the mortar or cement from setting. For this purpose, also, the centre should be lowered a trifle immediately after the key-stone is laid, in order that the stones may take their bearing before the mortar is set; otherwise the joints will open on the underside. The trusses, in centring, are placed at the distance of from 4 to 6 feet apart, according to their strength and the weight of the arch. Between every two trusses diagonal braces should be introduced to prevent lateral motion.

In order that the centre may be easily lowered, the frames, or trusses, should be placed upon wedge-formed sills, as is shown at d (Fig. 109). These are contrived so as to admit of the settling of the frame by driving the wedge d with a maul, or, in large centres, with a piece of timber mounted as a battering-ram. The operation of lowering a centre should be very slowly performed, in order that the parts of the arch may take their bearing uniformly. The wedge pieces, instead of being placed parallel with the truss, are sometimes made sufficiently long and laid through the arch, in a direction at right angles to that shown at Fig. 109. This method obviates the necessity of stationing men beneath the arch during the process of lowering; and was originally adopted with success soon after the occurrence of an accident, in lowering a centre, by which nine men were killed.

To give some idea of the manner of estimating the pressures, in order to select timber of the proper scantling, calculate the pressure (Art. 247) of the arch-stones from i to b (Fig. 109), and suppose half this pressure concentrated at a, and acting in the direction a f. Then, by the parallelogram of forces (Art. 71), the strain in the several pieces composing the frame bda may be computed. Again, calculate the pressure of that portion of the arch included between a and c, and consider half of it collected at b, and acting in a vertical direction; then, by the parallelogram of forces, the pressure on the beams bd and b dmay be found. Add the pressure of that portion of the arch which is included between i and b to half the weight of the centre, and consider this amount concentrated at d, and acting in a vertical direction; then, by constructing the parallelogram of forces, the pressure upon dj may be ascertained.

Joints Of The Arch-Stones

The strains having been obtained, the dimensions of the several pieces in the frames b a dand bed may be found by computation, as directed in the case of roof-trusses, from Arts. 226 to 229. The tie-beams b d, b d, if made of sufficient size to resist the compressive strain acting upon them from the load at b, will be more than large enough to resist the tensile strain upon them during the laying of the first part of the arch-stones below a and c.

248. - Arch-Stones: Joints. - In an arch, the arch-stones are so shaped that the joints between them are perpendicular to the curve of the arch, or to its tangent at the point at which the joint intersects the curve. In a circular arch, the joints tend toward the centre of the circle; in an elliptical arch, the joints may be found by the following process: To find the direction of the joints for an elliptical arch: a joint being wanted at a (Fig. no), draw lines from that point to the foci,f and f; bisect the angle f a f with the line a b; then a b will be the direction of the joint.

Joints Of The Arch Stones 141

Fig. 110.

Joints Of The Arch Stones 142

Fig. 111.

To find the direction of the joints for a parabolic arch; a joint being wanted at a (Fig. 111), draw a e at right angles to the axis eg; make eg equal to ce, and join a and g; draw ah at right angles to ag; then ah will be the direction of the joint. The direction of the joint from b is found in the same manner. The lines a g and b f are tangents to the curve at those points respectively; and any number of joints in the curve may be obtained by first ascertaining the tangents, and then drawing lines at right angles to them. (See Art. 462.)

Joints