Figure 2.1 Earth pressure at rest
                           (2.1)
Where  is effective vertical pressure and is unit weight of soil. Also, note that there are no shear stresses on the vertical and horizontal planes.
  If the wall is static－that is, if it does not move either to the right or to the left of its initial position－ the soil mass will be in a state of elastic equilibrium; that is, the horizontal strain is 0. The ratio of the effective horizontal stress to the vertical stress is called the coefficient of earth pressure at rest,  .
Since  , we have
                     (2.2)

Where   is effective horizontal pressure,  is an empirical coefficient, its value depends on the relative density of the soil, the process by which the deposit was formed, and its subsequent stress history.
For coarse-grained soils, the coefficient of earth pressure at rest can be estimated by the empirical relationship (Jaky, 1944)
                             (2.3)
Where   is drained friction angle. For fine-grained, normally consolidated soils, Massarsch (1979) suggested the following equation for  :

                     (2.4)

Where PI = plasticity index.
For overconsolidated clays, the coefficient of earth pressure at rest can be approximated as              (2.5)
Where OCR is overconsolidation ratio.
The magnitude of   in most soils ranges between 0.5 and 1.0, with perhaps higuer values for heavily overconsolidated clays.
Figure 2.1 shows the distribution of earth pressure at rest on a wall of height h, so
                                 (2.6)
If the groundwater table is located at a depth   below the ground surface, and there is no compensating water on the other side of the wall. For  , the total lateral earth pressure at rest can be given as  . However, for  , that is, below the groundwater table, the pressure on the wall can be found from the effective stress and pore water pressure components. Details see the book “Fundamemtals of Geotechnical Engineering”(Braja M. Das, 2005)

2.2.2 Rankine’s Earth Pressure Theory
Rankine (1857) investigated the stress conditions in soil at a state of plastic equilibrium. Figure 2.2 shows the Rankine’s active state and Rankine’s passive stateIt is bounded by a frictionless wall that extends to an infinite depth. The stress condition in the soil element can be represented by the Mohr’s circle II in Figure 2.2. However, if the wall is allowed to move away from the soil mass gradually, then the horizontal effective principal stress will decrease. Ultimately a state will be reached at which the stress condition in the soil element can be represented by the Mohr’s circle III, the state of plastic equilibrium, and failure of the soil will occur. This state is Rankine’s active state, and the pressure on the vertical plane is Rankine’s active earth pressure.

 
Figure 2.2 Rankine’s earth pressure

Following is the expression for Rankine’s active pressure. If vertical effective overburden pressure = , we have
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