PRESSURE OF EARTH AGAINST A WALL.
Theories of Earth Pressure.—The lateral pressure of a mass of earth against a retaining wall is affected by so many variable conditions that the determination of its actual value in a particular instance is practically impossible.
Several theories, based in each case upon certain ideal conditions, have been proposed, none of which are more than very rough approxi ntations to the conditions existing in such structures. These theo ries :-assume that the earth is composed of a grass of particles exerting friction upon each other but without cohesion, or that the pressure against the wall is caused by a wedge of earth which tends to slide upon a plane surface of rupture, a,s shown in Fig. 60. Formulas for the resultant thrust against the wall have been produced in accord ance with the various theories by several methods, they differ mainly in the direction given to the thrust upon the wall.
Coulomb's formula for computing the lateral thrust against a wall was proposed by Coulomb in 1773. Coulomb assumed that the thrust was caused by a prism of earth (B AC, Fig. 60) sliding upon any plane AC which produces the maximum thrust upon the wall. There is a certain slope (AD, Fig. 60) at which the material if loosely placed will stand. This is known as the natural slope, and the angle made by this slope with the horizontal as the angle of fric tion of the earth. On slopes steeper than the natural slope, there is a tendency for the earth to slide down, and if held by a wall, pressures are produced which depend upon the frictional resistance to sliding.
The thrust is assumed by Coulomb to he normal to the wall, and the pressure upon the plane of rupture to be inclined at the angle of friction to the normal to the plane.
Let It = height of wall; P= resultant pressure upon a unit length of wall; = pre sure upon the plane of rupture; G= weight of the wedge of earth; e =weight of earth per cubic foot; = angle of friction of earth; a =angle between the back of wall and plane of rupture.
For maximum value of P, and the plane of rupture bisects the angle between the hack of the wall and the natural slope.
Substituting this value, P varies as the square of h, and is therefore applied at a distance h,'3 above the base of the wall. This is the same in all of the theories.
Poncelet's Theory.—In 1840 Poncelet proposed to modify the method of Coulomb by making the thrust upon the wall act at the angle of friction with the normal to the wall.
Before the wall can he overturned about its toe (F, Fig. 61) the hack of the wall (A R) must he raised and slide upon the earth behind it, thus calling into play the friction of the earth upon the wall as a resistance. As the friction of earth upon a rough masonry wall is greater than that of earth upon earth, a film of earth would be carried with the wall and slide upon the earth behind and the angle of friction is usually taken as equal to the natural slope of the earth.
Let 0=the angle made by the back of the wall with the horizontal; i=the angle made by the earth surface with the horizontal.
Following the same procedure as in developing Coulomb's formula, we find the pressure against the wall, For a vertical wall and horizontal earth surface and which is the formula proposed by Poncelet.
IRankine's Theory. Rankine considered the earth to be made up of a homogeneous mass of particles, possessing frictional resistance to sliding over each other but without cohesion. Ile deduced for mulas for the pressure upon ideal plane sections through an unlimited mass of earth with plane upper surface, the earth being subject to no external force except its own weight, and determined the direction of the pressure from these assumptions.
Rankine found that the resultant pressure upon any vertical plane section through a bank of earth with plane upper surface is parallel to 1 he slope of the upper surface (see Fig. 62).
Let E= the pressure upon the vertical section; angle made by the inclination of the upper surface with the horizontal; is Rankine's formula for earth pressure. This pressure acts upon the vertical section at a distance S, 3 from its base, and makes an angle i with the horizontal.