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Where is unit weight of soil, and are cohesion and internal friction angle of soils, respectively.
For cohesionless soils, and
(2.8)
Rankine’s passive state is also illustrated in Figure 2.2. The reference wall is a frictionless wall that extends to an infinite depth. The initial stress condition on a soil element is represented by the Mohr’s circle I. At this time, failure of the soil will occur. This is referred to as Rankine’s passive state, the effective lateral earth pressure, which is the major principal stress, is called Rankine’s passive earth pressure. From Figure 2.2, it can be shown that
(2.9)
For cohesionless soils, , we have
(2.10)
Rankine’s earth pressure theory only considered the case of frictionless walls. In reality, retaining walls are rough, and shear forces develop between the face of the wall and the backfill. To understand the effect of wall friction on the failure surface, refered to the book “Fundamemtals of Geotechnical Engineering” (Braja M. Das, 2005)
2.2.3 Coulomb’s Earth Pressure Theory
More than 230 years ago, Coulomb (1773) presented a theory for active and passive earth pressures against retaining walls. In this theory, Coulomb assumed that the failure surface is a plane. The wall friction was taken into consideration. As shown in Figure 2.3.
(a) (b)
Figure 2.3 Coulomb’s earth pressure: (a) Coulomb’s active pressure ; (b) Coulomb’s passive pressure
Let AB be the back face of a reatining wall supporting a granular soil, the surface of which is constantly sloping at an angle with the horizontal. BC is a trial failure surface, and the probable failure wedge is ABC.
The force triangles for the wedge are shown in Figure 2.3. Based on the law of sins, we get
(2.11)
And
(2.12)
Where and are active pressure and passive pressure, respectively; is the effective weight of the soil wedge; the values of are constants, and is the only variable, which can be used to determine the critical value of for maximum and . We can obtain Coulomb’s active earth pressure as
(2.13)
Where is Coulomb’s active earth pressure coefficient, given by
(2.14)
And
(2.15)
Where is Coulomb’s passive earth pressure coefficient, given by