The variation in the values of a soil's strength parameters with drainage conditions has been discussed in Chapter 3 and it is important that the reader has an understanding of this phenomenon. A soil can exhibit shear resistance in one of three ways:
(i) due entirely to friction, its cohesive intercept = 0. The soil acts as a cohesionless soil,
(ii) Due entirely to cohesion, its angle of shearing resistance equals 0°. The soil acts as a cohesive soil,
(iii) a mixture of cohesive and frictional strength with both the cohesive intercept and the angle of shearing resistance having values above 0.0. The soil acts as a cohesive-frictional soil,
It can be seen therefore that the calculated value for the lateral pressure generated by the weight of a soil mass can be considerably in error if wrong values are assumed for the operative values of the cohesion and angle of shearing resistance of the soil. This aspect is considered later in this chapter but in the discussions in the early parts of this chapter the general symbols c and <f> are used. The reader should appreciate that these symbols can be exchanged for c' and <j>' when necessary.
Let us consider the simple case of a retaining wall with a vertical back (details of wall design and construction are given in Chapter 7) supporting a cohesionless soil with a horizontal surface (Fig. 6.1). Let the angle of shearing resistance of the soil be (f> and let its unit weight, 7, be of a constant value. Then the vertical stress acting at a point at depth h below the top of the wall will be equal to 7I1.
If the wall is allowed to yield, i.e. to move forward slightly, the soil is able to expand and there will be an immediate reduction in the value of lateral pressure at depth h, but if the wall is pushed slightly into the soil then the soil will tend to be compressed and there will be an increase in the value of the lateral pressure.
The above indicates that there are two possible modes of failure that can occur within the soil mass. If we assume that the value of the vertical pressure at depth h remains unchanged at 7I1 during these operations, then the minimum and maximum values of lateral earth pressure that will be achieved can be obtained from the Mohr circle diagram (Fig. 6.2).
The lateral pressure can reduce to a minimum value at which the stress circle is tangential to the strength envelope of the soil; this minimum value is known as the active earth pressure and equals Ka7h where Ka = the coefficient of active earth pressure. The lateral pressure can rise to a maximum value
(with the stress circle again tangential to the strength envelope) known as the passive earth pressure, which equals Kp7h where Kp = coefficient of passive earth pressure.
It can be seen from Fig. 6.2 that when considering active pressure the vertical pressure due to the soil weight, 7I1, is a major principal stress and that when considering passive pressure the vertical pressure due to the soil weight, 7h, is a minor principal stress.
The two major theories to estimate active and passive pressure values are those by Coulomb (1776) and by Rankine (1857). Both theories are very much in use today and both are described below.
6.3.1 Rankine's theory (soil surface horizontal)
Imagine a smooth, vertical retaining wall holding back a cohesionless soil with an angle of internal friction </>. The top of the soil is horizontal and level with the top of the wall. Consider a point in the soil at a depth h below the top of the wall (Fig. 6.3), assuming that the wall has yielded sufficiently to satisfy active earth pressure conditions. In the Mohr diagram:
<73 _ OA _ OC - AC _ OC - DC QC 1 — sin OB ~ OC + CB ~~ OC + DC + ~ DC ~ 1 +sin<ft
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