Conjugate Failure Planes Under Active State
When a backfill of cohesionless soil is under an active state of plastic equilibrium due to the stretching of the soil mass at every point in the mass, two failure planes called conjugate rupture planes are formed. These are further designated as the inner failure plane and the outer failure plane as shown in Fig. 19.1. These failure planes make angles of αi and α0 with the vertical. The equations for these angles may be written as (for a sloping backfill)

Conditions for the Use of Rankine's Formula
1. Wall should be vertical with a smooth pressure face.
2. When walls are inclined, it should not come in the way of the formation of the outer failure

plane. Figure 19.1 shows the formation of failure planes. Since the sloping face AB' of the retaining wall makes an angle αw greater than αo, the wall does not interfere with the formation of the outer failure plane. The plastic state exists within wedge ACC'.

The method of calculating the lateral pressure on AB' is as follows.

1. Apply Rankine's formula for the vertical section AB.
2. Combine Pa with Ws , the weight of soil within the wedge ABB', to give the resultant PR.

Let the resultant PR in this case make an angle δr with the normal to the face of the wall. Let
the maximum angle of wall friction be δm. If δr > δm, the soil slides along the face AB'of the wall.

Figure 19.1 Application of Rankine's active condition to gravity walls

Figure 19.2 Lateral earth pressure on cantilever walls under active condition

In such an eventuality, the Rankine formula is not recommended but the Coulomb formula may be

Conditions for the Use of Coulomb's Formula

1. The back of the wall must be plane or nearly plane.
2. Coulomb's formula may be applied under all other conditions where the surface of the wall is not smooth and where the soil slides along the surface.

In general the following recommendations may be made for the application of the Rankine or

Coulomb formula without the introduction of significant errors:

1. Use the Rankine formula for cantilever and counterfort walls.
2. Use the Coulomb formula for solid and semisolid gravity walls.

In the case of cantilever walls (Fig. 19.2), Pa is the active pressure acting on the vertical section AB passing through the heel of the wall. The pressure is parallel to the backfill surface and acts at a height H/3 from the base of the wall where H is the height of the section AB. The resultant pressure PR is obtained by combining the lateral pressure Pa with the weight of the soil Ws between the section AB and the wall.

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