A complete treatise on stone bridging would be of little practical value to the American engineer, and would occupy too much of the necessarily small space allowed here. The object in the present chapter is to give the manner of dimensioning stone arches of from ten to sixty feet span, and of proportioning retaining walls, piers, and abutments.

CONTRACTION OF THE WATER-WAY.

246. In building a bridge across a stream, we must be careful not to obstruct the water-way so as to prevent free passage to the highest floods. Regard must be had to this in fixing the size of the spans, and the thickness and number of the piers. By contracting the width of the stream the velocity is increased beneath the arches, the same amount of water being obliged to pass through a smaller space, and when the bottom is of such a nature as to yield to this action, there is danger of the foundation being undermined. If the form and size of the piers be so arranged as not to increase the velocity, such danger will be avoided and floods will pass without harm. In bridges crossing navigable streams, if the bottom is not destroyed the velocity may be made so great as to impede navigation.

247. The following table is from Gauthey, Construction des Ponts, showing the velocities which are just in equilibrium with the material composing the bottom of the stream.

248. If b represents the width of the natural water-way; c, that as reduced by the structure; V, the velocity of the stream in the natural state; then the augmented velocity is expressed by

where m is a constant quantity expressing the contraction which takes place in passing the narrow place, which, according to Du-Buat, is 1.09; but depending somewhat upon the form of the bridge piers; adopting which value, we have

Example.—Let the bottom be gravel, the width of the natural water-way one hundred feet, the velocity one foot per second: now for a gravel bottom the velocity must not exceed two feet per second, whence

which is the width of the contracted water-way; and 100 – 54½, or 45½ feet may be occupied by piers or other obstructions.

The amount of fall which the water suffers in passing the pier is found by the following formula, the notation being the same,

Thus the velocity being one foot per second, m being 1.09 and b = 100; also, c = 54½, we have

The velocity of a river is greatest at its surface and at the centre of the stream. In the same river the velocity is nearly as the square root of the depth; thus the surface velocity being known, that for any other depth may be easily found. The velocity of streams should always be noted at the times of the highest floods. For measuring the velocity of running water a bottle enough filled with water to maintain an upright position, with a small rod placed through the stopper having a red flag upon the upper end, answers very well. Velocities of undercurrents may also be measured by so loading the bottle as to cause it to float two, four, six, or ten feet below the surface.

249. There are three general forms which may be given to the intrados of a stone arch.

Semicircular, or one hundred and eighty degrees.

Segmental, less than one hundred and eighty degrees.

Basket handle, nearly elliptical, being formed by a number of circular curves.

Full centre (semicircular) arches offer the advantages of great solidity and ease of construction; but unless the springing lines are high, contract considerably the water-way.

Segmental arches give the freest passage to the water, are easily built, but throw a great horizontal strain upon the abutments.

The basket handle gives free passage to the water, when not too flat are very strong, are easily adjustable to different ratios between the span and the distance between grade and the spring line, and except making the centres, are easily built. Whatever the form of the arch, the line of arch springing should not be below high water.

The manner of tracing the full centre and segmental curves is too simple to need remark.

250. In tracing the basket handle curve, the following conditions must be observed:—

The number of arches composing the curve must not be less than three, nor more than eleven; and must be uneven. Perronet’s fine bridge of Neuilly, over the Seine at Paris, has eleven centres. In spans of sixty feet and under, it is unnecessary to use more than five centres.

251. The three centred curve is described as follows, fig. 119:—

Let A B represent the span, and c D the rise, with c as a centre and c A as radius, describe the quadrant A F E; make the angle A C F 60°. Parallel to F E draw D G, and parallel to F C draw G K. H is the centre, and A G the arc of the springing curve; also GD is the arc, and K the centre of the summit curve.

THE FIVE CENTRED CURVE.

252. The common construction of the five centred curve leaves the radii of the extreme curves to be assumed. The following method fixes all of the dimensions when the span and rise are given:—

Let c B be half the span and c D the rise.

Join D B.

Draw n K through n perpendicular to D B.

Make B a equal to c D.

Also c e to c a.

Draw e K′ o and K a m.

K H′ and K′ are the centres, and H′ m and H′ o the lines separating the several curves.

For spans of from twenty-five to one hundred feet, the five centred arch answers every purpose; making a strong and well proportioned structure.

THICKNESS OF VOUSSOIRS, (RING STONES).

253. The thickness of the voussoir, or arch stone, depends upon the form and size of the arch, the nature of the masonry, and the character of the stone. No authority gives more reliable results than Gauthey, who, for stone of average quality, with hammer dressed beds, laid in cement, gives the following proportions between the span and depth of key:—

For spans under six feet the depth should be thirteen inches.

From six to fifty feet, 13 inches plus 1
48 of the span.

From fifty to one hundred feet, 1
24 of the span.

For over one hundred feet, 1
24 of 100 plus 1
48 of the remainder.

Thus for a span of one hundred and ninety-six feet we have

or, 50 + 24 equal in all to seventy-four inches, or six feet and two inches; whence the following table:—

THICKNESS AND FORM OF ABUTMENTS.

254. The above depend upon the span and form of the arch, the height of the abutment, and character of the masonry.

Different methods of determining the thickness of an abutment have from time to time been given; several very correct rules have been arrived at, but difficult of application. The most simple rule is given by Hutton in the course of mathematics edited by Rutherford; it is as follows:—

Let A B, C D, fig. 120, be one half of the arch and A G F the abutment.

From the centre of gravity K of the arch, draw the vertical K L; then the weight of the arch in the direction K L will be to the horizontal thrust, as K L to L A. For the weight of the arch in the direction K L, the horizontal thrust L A, and the thrust K A will be as the three sides of the triangle K L, L A, K A; so that if m denotes the weight of the arch,

will be its force in the direction L A, and

its effect on the lever G A to overturn the wall, or cause it to revolve about the point F.

Again, the weight or area of the pier is as EF × FG, and therefore EF × FG × ½FG, or ½FG² × EF, is its effect upon the lever ½FG, to resist an overthrow. Now that the abutment and the arch shall be in equilibrium these two effects must be equal to each other; whence we must have

whence

The following table has been calculated for the use of builders and engineers, giving the thickness of abutments for different spans and heights.

256. The form of a bridge abutment will depend upon the locality and upon the use to which the bridge is to be put, whether used for a railroad, or for common travel; whether near a large city, or in a location where appearance need not be regarded. Where a river acts dangerously upon a shore, wing walls will be necessary. These wings may be curved or straight, and may be simply the abutment produced, or may be swung around into the bank at any required angle, until the winged abutment, as in figs. 121, 121 A, 121 B, becomes the U abutment, fig. 124; or by moving the walls, W and W, parallel to themselves, takes the form of the T abutment, fig. 122.

The curved wing, in fig. 121, being arched, requires a little less thickness, but at the same time is longer. B B, show the bridge seats. The slope of the wings may be battered with an inclined coping, or off-setted at each course. Wing walls, subjected to special strains or to particular currents of water, require positions and forms accordingly. In skew bridges, as in Chap. V., the wing, at the acute angle, is longer and inclines less from the face of the abutment than that at the obtuse angle. The more the wing departs from the face line and swings round into the slope, the greater the thrust becomes upon it, as the centre of pressure is raised; the thrust becomes a maximum when the wing is inclined from forty-five to seventy degrees from the face of the abutment. The body of an abutment, as well as any other retaining wall, may be much stronger by giving it a trapezoidal instead of a rectangular section, as the resisting leverage is thereby much increased. Abutments may be to advantage buttressed in order to resist special strains, as in case of the arches or braces of wooden bridges.

257. Railroad abutments except for a double track, require but little breadth on top, except where the truss itself rests. The common T abutment originated by Captain John Childe, and now in very extensive use, seems to fulfil any requirement of a good abutment, see fig. 122, page 242. B B is the bridge seat, and the mass T T takes the place of wings. The difference of level of the top and of the bridge seat depends upon the difference between the height of the bearing of the lower chord of the bridge, and grade. The line of contact between the earth and the wall is shown by s s′ s″ s‴. The length of the top of the masonry is found thus. Suppose the slope to be one and one half to one, and the whole height thirty feet, the whole horizontal length of slope is then forty-five feet; from this we take the sum of the horizontal distances, s s′ and s′ s″, and suppose these to be, respectively, six and eight feet, we have the whole operation thus:—

It may be advisable in very high abutments to lighten the masonry by an arched opening as in fig. 123. The walls, also, of the U abutment (see fig. 124), when large, may be pierced with arches to save masonry.

Probably the cheapest mode of bringing a bridge to the embankment is that shown in fig. 125; A being the bridge seat for the main truss, and B that for the trussed girder.

258. The thickness of a pier may be considered either as depending upon the weight of the superstructure, or as resisting the thrust of arches or braces. For the first requirement, very little thickness would suffice; for the second, it may require to be considerable. The objection to thick piers is the expense, and the contracting too much the water-way; the benefit, a large bearing surface, and in stone bridges where there are several continuous spans, a saving of centring; as where the piers are not able to resist the thrust of the arches, they must all be carried up at once.

259. Piers supporting truss bridges, require very little thickness provided a good foundation is obtained. The following table shows the sufficient dimensions for the piers of wooden or iron trussed bridges, when the masonry is good. (See First Class Masonry, specification, Chap. IV.) From ten to twenty feet in height the batter is assumed at one twelfth; from twenty to fifty feet in height at one twenty-fourth.

260. Upon the form of the up-stream end of the pier, or the starling, depends, in a considerable degree, the contraction of the water-way. In sluggish water the form is not of much importance, but in swift flowing rivers a great deal depends upon the choice. The forms in use are the rectangle, the rectangle terminated by right-lined triangles, and the same terminated by curved-lined triangles, and finally the ellipse.

The latter is that which causes the least disturbance to the water, but is also the most costly.

The effect of gyration at the shoulder, deserves notice, as it may be the cause of the ruin of the foundation when the bottom is of yielding material.

River beds being porous, springs work up through them with a force equal to the whole depth of water; and whenever there is a means of escape for such, its pressure will act upwards against any structure that comes within its reach; and if four or five feet deep, is capable of moving enormous weights. Such springs gave a great deal of trouble at the foundation of the United States Dry Dock, at Brooklyn, N. Y. When checked in one place they burst up in another, and to proceed with the work it was necessary to allow them a passage through which to flow.

261. However proper it may be to give to piers the proper form to cause as little contraction as possible to the water, it is no less necessary to give them strength to oppose the shocks to which they are subject from floating ice, timber and shipping. The best method of breaking up ice, when it comes in large masses, is by inclining the front of the pier, as shown in fig. 126. The angle of the front being inclined from 30° to 50°. The ice running up this slope breaks by its own weight, and falls off on either side.

The foundations of piers may be protected by sheet piling, (see chap. XII.,) or the bottom, if soft, may be dredged out for a few feet and filled in with loose rock.

The form of the down-stream end is not of so much importance as of the upper one, but deserves consideration; as when the water is swift or the bottom soft and yielding, the eddies caused by sharp angles wear upon the soil in a dangerous manner.