Determination of Initial Total Stresses.

A homogeneous soil deposit with a horizontal ground surface is shown in Figure 3.9, and the vertical stress at depth z is desired. A column of soil of area A as shown may be considered. Static equilibrium in a vertical direction requires that the weight of the column W plus any shear forces on the sides to equal the force on the bottom. Since the soil deposit is homogeneous and extends indefinitely in the horizontal direction, the stresses on all vertical planes are the same. Thus, if any shear exists on vertical planes, the shear on the outside of one face must be identical to the shear on the outside of the opposite face, so the shear forces are in equilibrium. Actually, if the soil

Figure 3.9 Equilibrium of vertical forces for a vertical prism of soil.

deposit is formed by uniform deposition over the surface, no shearing stresses on vertical planes will occur. Thus, for force equilibrium in the vertical direction
where is the total stress on the base. The stress is then
 As a matter of convenience, substitute


where v is the volume of the column of soils and  is the total unit weight (total weight per unit volume). After inserting Eq. 3.36 into Eq. 3.35, the following equation results
Equation 3.37 is valid only if   is constant with depth. For real (nonhomo-geneous) soils, the soil may be considered homogeneous on a microscopic scale, that is, in differential elements, so
and

Equation 3.39 is valid for any soils with a horizontal ground surface. The total unit weight  can be found for soil samples by measuring the weight and volume of the soil samples, but  is computed more conveniently
from other parameters that are measured for other purposes.
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