## Stability of a finite slope based on method of slices

Bromhead5 defines the factor of safety used in limit equilibrium methods as the ratio of the mobilized shear strength to the actual shear strength available. This is akin to applying a partial factor to material strength and hence those Design Approaches that apply partial factors to material properties are highly suited to the solution of slope stability problems.

Methods for the limit equilibrium analysis of slopes range from simple translational sliding along a flat plane (as discussed in Section 9.4) to movement along complex shaped sliding surfaces involving curved and planar surfaces. The simplest forms of curved surface involve circular arcs and the discussion below concentrates on such surfaces, which typically compare the overturning moment MO about a point of rotation to the restoring moment MR: Mr

In order to take into account varying conditions along the slip surface and within the sliding mass, it is usual to split the sliding mass into slices and consider the stability of each slice in turn (see Figure 9.7). The results for each slice are summed to give both the overall values for the slope and the factor by which the soil strength needs to be reduced in order to induce failure.

In drained (effective stress) analyses, the available shear strength along the failure plane is a function of the effective normal stress and any effective cohesion acting on that plane. In undrained (total stress) analyses, the available shear strength is a function of the undrained strength alone, which simplifies the analysis considerably.

Figure 9.7 shows a typical slope with the key features of a circular slip analysis indicated. The overturning moment MO is defined as:

i where Wi is the self-weight of slice 'i'; Qi is any applied surcharge acting on that slice; and xi is the slice's lever arm about the point of rotation, O. In traditional calculations, the applied surcharge Qi is included in the weight term Wi. However, because Eurocode 7 applies different partial factors to permanent and variable actions (i.e. to self-weight and applied surcharge, respectively, in this instance), we have separated the terms in this book.

It should be noted that x can take either positive or negative values and so MO has components which are either unfavourable, i.e. increase the overturning moment (positive x); or favourable, i.e. decrease the overturning moment (negative x).

9.5.1 Undrained analysis based on total stresses

In total stress analyses, the restoring moment MR is defined as: M = r xY\c . x l.}

R L_J y u,i ij i where r is the radius of the slip circle; li is the length of the slip surface beneath 'i'; and cu,i is the undrained shear strength along the base of that slice.

The factor of safety F is defined as:

i where aj is the angle between the base of the slice and the horizontal, i.e.: a = sin-1 (xJr)

and the other terms are as defined above. This is known as the 'Conventional Method' of analysis.

9.5.2 Eurocode 7 implementation of the Conventional Method

As discussed in Chapter 6, verification of strength according to Eurocode 7 involves demonstrating that design effects of actions Ed do not exceed the corresponding design resistance Rd. i.e.:

In slope stability analysis, the overturning moment MO is an action effect and the restoring moment MR is the resistance to that effect. Hence Eurocode 7 requires the design of slopes and embankments to satisfy:

where Wdi = the design self-weight of slice 'i'; Qdi = any imposed surcharge acting on that slice; cudi = the design undrained shear strength along the base of the slice; and the other terms are as defined above.

This equation can be re-written in terms of characteristic parameters, as follows:

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