3.2.2 Conventional earth pressure theory
When soil is removed from the front of an embedded retaining wall，the wall tends to move into the excavation.This will result in a reduction in the lateral stress in the ground behind the wall，eventually bringing it to the active condition in which the ground is at failure with the horizontal effective stress as small as it can be for the effective overburden pressure.In the ground that remains in front of the wall below formation level，the horizontal effective stress at failure is as large as it can be for the effective overburden pressure.
Approximations to these limiting pressures may be calculated by considering either the stresses in a zone of ground at failure，or the equilibrium of an assumed sliding wedge.The first approach，following Rankine（1857），is based on a stress state that can be in equilibrium without exceeding the limiting strength of the ground.In a uniform deposit，the limiting ratio of horizontal to vertical effective stresses is constant.That is，if the vertical effective stress increases linearly with depth，so will the limiting horizontal effective stress.The limits calculated in this way ensure stability，but may be unnecessarily severe，and are inherently safe.（assuming that the correct boundary conditions，ground strength parameters and pore pressures have been identified）
In the second approach，following Coulomb（1776），the force that must be exerted by a retaining wall to prevent a wedge of soil from sliding down an assumed slip surface is determined，and is then usually assumed to arise from a stress that increases linearly with depth.The limits obtained will prevent sliding along the slip surface assumed in the calculation，but may not be sufficient to prevent failure from occurring in some other unidentified mechanisms，so they could be unsafe.
The results obtained from Rankine's and Coulomb's earth pressure theories are identical under the same conditions（smooth wall surfaces，level grounds，and homogeneous cohesionless soils）though the two theories are quite differently based（Figures 3.3 and 3.4）.The actual conditions，however，may not conform to the hypothetical terms：the wall surface may be rough and the ground surface in back of the wall may be of irregular shape with certain load.Rankine's theory can hardly apply under these conditions.Coulomb's theory can cope with these complicated conditions，though it is difficult to obtain a theoretical solution.However，experience overwhelmingly demon⁃strates that these more recent theories give accurate values of the limiting lateral stresses for use as a basis for design，assuming that variation in pore water pressures and strength properties，etc.in the ground around the wall have been identified and taken into account.
Figure 3.3 Rankine plastic equilibrium for a frictionless wall or soil interface translating horizontally
Figure 3.4 Coulomb's method to calculate the limiting active force for a frictionless wall or soil interface translating horizontally
In summary，the conventional approach to the design of an embedded retaining wall is something of a hybrid，with earth pressure coefficients derived using a mechanism⁃based approach to take account of factors such as wall interface friction，the profile of the retained ground surface and the presence of line loads used in an equilibrium analysis of the wall.