The natural slope for the earth is the slope at which the surface of the material will stand when clumped into piles, the frictional resistance keeping the surface layer from sliding or rolling down the slope. The angle of sliding friction of a wedge of earth upon an earth surface may not be the same as the inclination of the natural slope. Values of sliding friction as determined by experiment vary considerably for t he same material, and it is possible that much of the variation is due to the methods of testing rather than to differences in the materials. The natural slope of a particular material may usually be approximately determined without difficulty and its use instead of the angle of sliding friction would ordinarily he safe. Table 1I1 gives approximate values of the angle of internal friction, the natural slopes and weights of various materials commonly tnet in construction.
Surcharged formulas for earth pressure already given assume the earth to carry only its own weight and the upper surface to slope from the top of the wall. When the earth behind the wall carries a load upon its surface, as when supporting a railway track or a pile of material of any sort, the pressure against the wall is increased uniformly over its entire depth.
If u+ is the weight of the load per unit area of earth surface, Formula (1) becomes, 125. Graphical Method.—When the surface of earth is irregular or broken, the formulas do not apply, although it is usually possible to approximate the plane surfaces with sufficient accuracy. Graph ical determination of earth pressures may be made when the slope of the surface is not too great. When the surface slope is near the
natural slope for the material, these methods cannot be used.
A graphical method is shown in Fig. 63. OA is the back of a wall, and ABC D, etc., the upper surface of the earth resting against it. Divide the earth into a number of prisms by the lines OB, OC, etc. On the line na lay off on some convenient scale ab, bc, etc., equal respectively to the weights of the prisms OA B, OBC, OCD, etc.
From a draw the lines etc., making the angle of friction (0) with the normals to OB, OC, 01), etc., respectively. From the points b, c, d, etc., draw the lines etc., making the angle of friction (0) with the back of the wall (OA), to intersection with the lines etc., respectively. The lengths ddi, etc., will then represent, on the scale to which the weights were laid off, the thrusts of the prisms between the back of the wall and the planes DB, OC, etc., respectively.
In the figure, eei is the maximum thrust, caused by the prism between OA and OT, showing OE to be the plane of rupture. This resultant thrust will net at a distance h/3 from the base of the wall, at the angle of friction with the normal to the wall.
Detailed discussions of methods of determining earth pressures are given in "Retaining Walls for Earth" by M. A. Ilowe, New York, 1896, and in "Practical Designing of Retaining Walls," by Wm. Cain, New York, 191.1. An interesting paper by E. P. Goodrich in Trans actions, .\nncrican Society .,f ('ivil Engineers. December, 1904, gives results of experiments for determination of internal friction and lateral pressure of earth.