Figure  shows a smooth, vertical wall supporting a homogeneous soil mass under undrained condition. Using the limit equilibrium method, we will assume, for the active state, that a slip plane is formed at an angle u to the horizontal. The forces on the soil wedge are shown in Figure 15.11. Using static equilibrium, we obtain the sum of the forces along the slip plane:

Equation  then yields

We are using P rather than Pa because Pa is the limiting value. To fi nd the maximum active lateral earth force, we differentiate P with respect to u and set the result equal to zero, giving

The solution is u =ua = 45⁰ .
By substituting u = 45⁰ into the above equation for P, we get the maximum active lateral earth force as

If we assume a uniform distribution of stresses on the slip plane, then the active lateral stress is

Let us examine Equation. If (sx)a = 0, for example, when you make an excavation, then solving for z from Equation  gives

 

 

Depth zcr is the depth at which tension cracks would extend into the soil (Figure). If the tension crack is fi lled with water, the critical depth can extend to

In addition, the soil in the vicinity of the crack is softened and a hydrostatic pressure, gwz9cr, is imposed on the wall. Often, the critical depth of water-fi lled tension cracks in overconsolidated clays is greater than the wall height. For example, if su 5 80 kPa, and g 5 gsat 5 18 kN/m3, then z'cr 5 19.5 m. A wall height
equivalent to the depth of the tension crack of 19.5 m is substantial. This is substantially more than the average height of the Great Wall of China, which is about 7.6 m. When tension cracks occur, they modify the slip plane, as shown in Figure 15.12; no shearing resistance is available over the length of the slip plane above the depth of the tension cracks. For an unsupported excavation, the active lateral force is also zero. From Equation, we get

 

and, solving for Ho, we obtain

If the excavation is fi lled with water, then

We have two possible unsupported depths, as given by Equations. The correct solution lies somewhere between these critical depths. In design practice, a value of

is used for unsupported excavation in fi ne-grained soils. If the excavation is filled with water

The passive lateral earth force for a total stress analysis, following a procedure similar to that for the active state above, can be written as

and the passive lateral pressure is

We can write Equations  using apparent active and passive lateral earth pressures for the undrained condition as

 

where Kau and Kpu are the undrained active and passive lateral earth pressure coeffi cients. In our case, for a smooth wall supporting a soil mass with a horizontal surface, Kau 5 Kpu 5 2. Walls that are embedded in fi ne-grained soils may be subjected to an adhesive stress (sw) at the wall face. The adhesive stress is analogous to a wall–soil interface friction for an effective stress analysis. The undrained lateral earth pressure coeffi cients are modifi ed to account for adhesive stress as

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