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CHAPTER 9
Stability of Slopes
9.1Introduction
Gravitational and seepage forces tend to cause instability in natural slopes,in slopes fo rmed by excavation and in the slopes of embankments and earth dams.The most important types of slope failure are illustrated in Fig.9.1.In rotational slips the shape of the failure surface in section may be a circular arc or a non-circular curve ．In general ，circular slips are associated with homogeneous soil conditions and non-circular slips with non-homogeneous conditions ．Translational and compound slips occur where the form of the failure surface is influenced by the presence of an adjacent stratum of significantlydifferent strength ．
Translational slips tend to occur where the adjacent stratum is at a relatively shallow depth below the surface of the slope:the failure surface tends to be plane and roughly parallel to the pound slips usually occur where the adjacent stratum is at greater depth ，the failure surface consisting of curved and plane sections
．
In practice,limiting equilibrium methods are used in the analysis of slope stability.It is considered that failure is on the point of occurring along an assumed or a known failure surface ．The shear strength required to maintain a condition of limiting equilibrium is compared with the available shear strength of the soil ，giving the average factor of safety along the failure surface ．The problem is considered in two dimensions ，conditions of plane strain being assumed ．It has been shown that a two-dimensional analysis gives a conservative result for a failure on a three-dimensional(dish-shaped)surface ．
9.2Analysis for the Case of φu =0
This analysis,in terms of total stress ，covers the case of a fully saturated clay under undrained conditions,i.e.For the condition immediately after construction ．Only moment equilibrium is considered in the analysis ．In section,the potential
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failure surface is assumed to be a circular arc.A trial failure surface(centre O ，radius r and length L a )is shown in Fig.9.2.Potential instability is due to the total weight of the soil mass(W per unit Length)above the failure surface ．For equilibrium the shear strength which must be mobilized along the failure surface is expressed
as
where F is the factor of safety with respect to shear strength ．Equating moments about O
：
Therefore
(9.1)
The moments of any additional forces must be taken into account ．In the event of a tension crack developing ，as shown in Fig.9.2，the arc length L a is shortened and a hydrostatic force will act normal to the crack if the crack fills with water ．It is necessary to analyze the slope for a number of trial failure surfaces in order that the minimum factor of safety can be determined ．
Based on the principle of geometric similarity ，Taylor[9.9]published stability coefficients for the analysis of homogeneous slopes in terms of total stress ．For a slope of height H the stability coefficient (N s )for the failure surface along which the factor of safety is a minimum
is
(9.2)
For the case of φu =0，values of N s can be obtained from Fig.9.3.The coefficient N s depends on the slope angle βand the depth factor D ，where DH is the depth to a firm stratum ．
Gibson and Morgenstern [9.3]published stability coefficients for slopes in normally consolidated clays in which the undrained strength c u (φu =0)varies linearly with depth ．
Example 9.1
A 45°slope is excavated to a depth of 8m in a deep layer of saturated clay of
