There are three standard methods to estimate earth pressure in active and passive conditions: Rankine's, Coulomb's, and Log-spiral Methods. Rankine's method is the simplest, with certain analysis assumptions. We can derive it from Mohr's circle, as shown in Fig. 1, for cohesionless soil (c = 0).

To simplify Coulomb's earth pressure theory, Rankine proposed some assumptions for his analysis:

__Assumptions__

Soil is homogeneous and isotropic

The ground surface is plane (can be inclining)

Neglect friction between wall and soil

Neglect friction between soil at the failure plane

__Formulas__

For level ground surface, the Rankine's active and passive pressure of cohesionless soil can be computed from:

For sloping backfill, Rankine's active and passive pressure coefficients are written as

__Failure mechanisms__

- Active soil failure is achieved when the lateral movement of the wall (out of the soil) causes a reduction in lateral pressure to the failure envelope.

- Conversely, passive failure occurs when the lateral movement of the wall (into the soil) causes an increase in lateral pressure to the failure envelope.

__Shear failure plane__

To achieve either an active or passive failure state, a certain magnitude of horizontal movement of the wall must be satisfied, as shown in the tables below:

Rankine's method can be used to design retaining walls as it provides a conservative approach for both active and passive pressures. However, it's more practical to use Coulomb's method to predict active earth pressure behind the wall and use Rankine's method to predict earth pressure in front of the wall.

DESIGN GUIDE

Rankine's theory only applies to walls with a vertical rear face. The pressure also acts on the wall at the same angle as the ground surface.

2. Coulomb's method is highly unsafe when used to compute passive pressure; the value can be two times greater than predicted by the log-spiral method and several times Rankine's.

REFERENCE

DP Coduto, Foundation Design: Principles and Practices, 52nd edition Prentice Hall, 2001

Pile Buck Inc., Steel Sheet Piling Design Manual, 1987