The actually computed weight of a unit-section of the abutment is determined by multiplying the sum of a/file' and ch by the base cd. Since this masonry is assumed to weigh 160 pounds per cubic foot, the product of these scaled measured at the scale of inch equals one foot (which was the scale adopted for the original drawing), shows that the section one foot thick has a weight of 19,500 pounds. Laying off this weight from the point k, and laying off the pressure for the first condition of loading, 16,350 pounds, at the scale of 5,000 pounds per inch, and forming a parallelogram on these two lines, we have the resultant of 31,350 pounds as the pressure on the base of the abutment, that pressure passing through the point in.
The intersection of the weight of the abutment with the line of pressure for the third condition of loading, is a little below the point k; and we similarly form a parallelogram which shows a resulting pressure of 33,600 pounds, passing through the base at the point n. It is usually required that such a line of pressure shall pass through the middle third of the abutment; but there are other conditions which may justify the design, even when the line of pressure passes a little outside of the middle third.
The point a is 2.S5 feet from the point e. According to the theory of pressures enunciated in Article 405, it may be considered that the pressure is maximum at the point e, and that it extends backward toward the point d for a distance of three times en, or a distance of 8.55 feet. This would give an average pressure of 3,930 pounds per square foot, or, since the pressure at the toe is twice the average pressure, 7,860 pounds per square foot on the toe. Such a pressure might or might not be greater than the subsoil could endure without yielding. Since this pressure is equivalent to 'about 55 pounds per square inch, there should be no danger that the masonry itself would Mil; and, if the subsoil is rock or even a hard, firm clay, there will be no danger in trusting such a pressure on it.
Another very large item of safety which has been utterly ignored, but which would unquestionably be present, is the pressure of the earth back of the abutment. The effect of the back-pressure of the earth would be to make the line which represents the resultant pres sure on the subsoil more nearly vertical, and to make it pass much more nearly through the center of the base ed. This would very much
reduce the intensity of pressure near the point c, and would reduce very materially the imit-pressure on the subsoil. Cases, of course, are conceivable, in which there might be no back-pressure of earth against the rear of the abutment. In such cases, the ability of the subsoil to withstand the unit-pressure at the rear toe of the abutment (near the point c) must be more carefully considered. In order that the investigation shall be complete, it should lie numerically deter mined whether the lower pressure, 31,350 pounds, passing through the point might produce a greater intensity of pressure at the point e than the larger pressure passing through the point a.
416. Vzirious Forms of Abutments. The abutment described above is the general form which is adopted very frequently. The front face ed is made with a batter of one in twelve. The line be slopes backward from the arch on an angle which is practically the con tinuation of the extrados of the arch. The total thickness of the abut ment de must be such that the line of pressure will come nearly, if not quite, within the middle third. The line ea generally has a con siderable slope, as is illustrated. When the subsoil is very soft, so that the area of the base is necessarily very great, the abutment is sometimes made hollow, with the idea of having an abutment with a very large area of base, but which does not require the full weight of so much masonry to hold it down; and therefore economy is sought in the reduction of the amount of masonry. Since such a hollow abutment would require a better class of masonry than could be used for a solid block of masonry, it is seldom that there is any economy in such methods. 'Since the abutment of an arch invariably must with stand a very great lateral thrust from the arch, there is never any danger that the resultant pressure of the abutment on the subsoil will approach the front toe of the arch, as is the ease in the abutment of a steel bridge, which has little or no lateral pressure from the bridge, but which is usually subjected to the pressure of the earth behind it. These questions have already been taken up under the subject of abutments for truss bridges, in Part II.