Assume a very simple numerical case, as in Fig. 67. The weight of the wall and its line of action are very readily determined with accuracy. The base of the wall has been made 4/10 of the height, or 7.2 feet. The batter of the outer face is at the rate of 1 in 5, or is 3.6 feet in the total height of the wall, leaving 3.6 feet as the thickness at the top. The area of the cross-section = 1/2 (3.6 + 7.2) 18 = 97.2 square feet. On the basis that this masonry weighs 140 pounds per cubic foot, a section of this wall one foot long will weigh 13,608 pounds. To find the line of application of the weight we must find the center of gravity of the trapezoid, and for this purpose we may divide the trapezoid into a rectangle and a right-angled triangle. The rectangle has an area of 64.8 square feet, and its center of gravity is 1 8 feet from the rear face. The center of gravity of the triangle (whose area equals 1/2 X 18 X 3.6 = 32.4 square feet) is at one-third of the base of the triangle from its right-hand vertical edge, or at a distance of 4.8 feet from the rear of the wall. The center of gravity of the trapezoid is then found numerically as follows:

64.8 X 1.8 = 116.64

32.4/97.2 x 4.8 = 155.52/272.16 272.16 97.2 = 2.80 feet, which is the distance of the center of gravity of the trapezoid from the rear face of the wall. The pressure of the earth on the rear wall, as stated above, is very uncertain; a value for it has already been computed (in the example in section 225) as 5,400 pounds. This value is probably excessive, except under the most unfavorable conditions. The point of application of the resultant of this pressure, as well as the direction of that resultant, is also uncertain, and has been the subject of much theoretical controversy. If the soil were merely a liquid which had no internal friction, there would be no uncertainty. In this case, the point of application of the pressure would be at one-third the height of the wall from the base, and its direction would be perpendicular to the rear face of the wall. This is the most unfavorable condition for stability which could be assumed; and on this account, calculations are sometimes made on this basis, with the knowledge that if the wall is stable under these most unfavorable conditions, it will certainly be stable no matter what the real conditions may be. -On this basis we have the resultant pressure against the rear of the wall as indicated by the arrow in Fig. 67. The resultant pressure on the«base of the wall is therefore a line the direction of which is indicated by the diagonal of a parallelogram whose two sides are parallel to the two forces, the sides being proportional to those forces. The amount of this pressure equals the square root of the sum of the squares of 5,400 and 13,608, or 14,640 pounds. The intersection of this line of pressure with the base is evidently at a distance from the intersection of the line of vertical pressure, equal to;

That point is therefore 5.18 feet from the rear of the wall, or 2.02 feet from the toe. This point represents the center of pressure of the pressure on the subsoil. The pressure is most intense at the toe of the wall, and is there assumed to be twice as intense as the average pressure. It is also assumed that the pressure diminishes toward the rear, until, at a distance back from the center of pressure equal to twice the distance from the center of pressure to the toe, the pressure is zero. This would mean that the pressure varies as the ordinates of a triangle (as illustrated in Fig. 68), the triangle having a base of 3 X 2.02 = 6.06 feet. The average pressure would equal 14,640 ÷ 6.06 = 2,415 pounds per square foot. The maximum pressure at the toe would therefore equal twice this average pressure, or 4,830 pounds per square foot, or about 34 pounds per square inch. This unit-pressure is so far within that allowable for stone masonry, that there is no danger of the crushing of the masonry at the toe.

The pressure on the subsoil, which is less than 2 1/2 tons per square foot, is less than that usually allowable on a good subsoil. There is therefore but little danger that the subsoil will be crushed and that the wall will tip over bodily on account of the failure of the subsoil. Since the line of pressure is likewise two feet back of the toe of the wall, there is no danger that the wall will tip over around its toe. The accuracy with which these calculations have been carried out should not lead to the idea that the pressures will necessarily be exactly as stated, since the calculations are based on assumptions which are at the best very doubtful, but which, as previously stated, are probably excessively safe.

Fig. 67. Resultant Pressure of Retaining Wall.

The form chosen for this wall is also so simple that a purely numerical calculation was the easiest and most satisfactory method. If the shape of the wall had been more irregular, it would have been easier to adopt the graphical method for the determination both of the center of gravity of the wall and of the resultant pressure on the subsoil. For instance, if the rear face of the wall had been inclined, the line of pressure would have been drawn perpendicular to the rear face and through a point at one-third the height of the wall. The position of the center of gravity of the wall would have been determined by the purely graphical method of determining the center of gravity of a trapezoid; and then the amount, direction, and intersection of the resultant with the base of the wall would have been determined by purely graphical methods.