Handbook of Railroad Construction; For the use of American engineers. / Containing the necessary rules, tables, and formulæ for the location, construction, equipment, and management of railroads, as built in the United States.

by George L. Vose

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:—

Fig. 120.

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,

LA
KL × m,

will be its force in the direction L A, and

LA
KL × GA × m

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

½FG² × EF = LA
RL × GA × m;

whence

FG = √(2GA × LA
EF × RL × m).

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

255. THICKNESS OF RECTANGULAR ABUTMENTS.
	Semicircular arch.				Basket-handle arch.
	The height being.
Span.	5	8	10	15	5	8	10	15
6	3	3	3¼	3½	3½	4	4	5
8	3½	4	4¼	4½	4	4½	5¼	6
10	4	5	5	5	4½	5¾	6¼	7
15	4½	5¼	5½	6	5	6½	7¼	8
20	5	5½	6	7	6	7¼	8	9
25	5½	6	6½	7¼	7	7½	8¼	9½
30	6	7	8	8½	8	8½	9¼	10
35	6½	7	8¼	9	9	9¼	10	11
40	7	7½	8¾	9½	9½	10	11	12
45	7½	8¼	9¼	10	10	10¾	11½	12½
50	8	9	10	11	10¼	11½	12¼	13

Fig. 121.       Fig. 121 B.

Fig. 121 A.

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.

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.

Fig. 123.

Fig. 124.

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:—

30 × 1½ – 6 + 8 = 45 – 14 = 31 feet.

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.

Fig. 125.