Theories of Lateral Pressure

For an investigation as to the reliability of theoretical formulas, see § 998-1013.

Coulomb's Formula. If we assume (1) that the earth surface is horizontal, i.e., that 8 = 0, (2) that the back of the wall is vertical, i.e., that 0 = 90°, and (3) that the resultant earth pres sure is normal to the back of the wall, i.e., that z = 0, then equa tion 3 becomes which is the well-known expression first deduced by Coulomb in 1773.

Rankine's Formulas. If we assume that the resultant earth pressure makes an angle with the normal to the back of the wall equal to the angle of repose of earth on earth, that is, if we assume that the angle of friction between the earth and the back of the wall is the same as that of earth on earth, then z = 0, and equation 3 becomes which is Rankine's formula for the pressure against a wall having an inclined back.

If we assume further that the back of the wall is vertical, and also that the line of action of the resultant lateral pressure of the earth is parallel to the surface of the earth, then 0 = 90° and hence z = 8, and equation 3 becomes which is Rankine's general formula for the pressure against a vertical wall carrying a surcharge at an angle 8.

If

8 = 0, as is usually assumed, then which is Rankine's formula for a surcharge at the angle of repose.

Weyrauch's Formula. If it be assumed that the earth pressure is normal to the back of the wall, then z = 0, and equation 3 becomes which is one of the formulas proposed by Weyrauch in 1878. Poncelet's Formula. If it be assumed that z s 0, d — 0, and 0 a 90°, then equation 3 becomes which is the expression deduced by Poncelet in 1840.

Theories for the Point of Application of the Pressure. In

all theories, the amount of the lateral pressure is given by equation 3, page 492, or by an equation easily derived therefrom; and since by that formula the pressure varies as h', i.e., as the square of the vertical height of the wall, it is always assumed in theoretical in vestigations that the lateral pressure of earth follows the law of liquid pressure and that therefore the point of application is h from the bottom of the wall. For a comparison between theory and ex periment on this point, see $ 1001-8.

Theories for the Direction of the Pressure.

Equation 3,

page 492, gives the maximum lateral pressure in terms of z, the un known angle between the resultant pressure and the normal to the back of the wall. The real value of z can not be determined theo retically; and hence different investigators have assumed different values for this angle.

It seems reasonable to assume that the true value of z must lie between and the angle of friction of the earth against the back a the wall; but the angle of friction against the rough back of a stone block wall is not known, and hence it is usual to assume values of z between 0 and 0. For a level bank of earth, i.e., for S = 0, the value of E is less for z = 0 than for z = 0; but for a surcharge, i.e., for large values of 6, E is larger for z = 0 than for z = 0.

Usually the advocates of the theory of the prism of maximum thrust have assumed that the resultant pressure is normal to the wall, i.e., that z in equation 3 is equal to zero ; and usually those who use the principle of conjugate pressures have assumed that the resultant pressure is parallel to the surface of the supported earth, i.e., that z in equation 3 is equal to — 0 - d.

Angle of Repose.

To apply any of the preceding formulas for the lateral thrust of earth, it is necessary to know ˘, the angle of repose of the earth. This angle may be determined as follows: Thoroughly pulverize a mass of earth so as to destroy all cohesion between its particles; and then slowly pour it vertically upon a horizontal surface, thus forming a cone. The angle of the sides of this cone with the horizontal is 0, the angle of repose or the (Ingle of natural slope. The particles of earth on such a slope are held in equilibrium by the force of gravity and by friction.

Table 75 gives rough average values of the angle of repose and also of the weight of various kinds of earth. Slight varia tions in the amount of moisture make great differences in the value of the angle of repose and of the weight. The results in Table 75 are about those usually given in discussions of the theory of the stability of retaining walls; but it will presently be shown that any such results are not of much value.