Limit equilibrium and the factor of safety

A limit-equilibrium analysis asks a single question of a potential slip surface: by how much would the shear strength of the ground have to fall for the mass above it to be on the point of sliding? The answer is the factor of safety. OGR Slip2D computes it with the method of slices, which divides the sliding mass into vertical slices; the problem is then statically indeterminate, and each method of slices closes it with its own assumption about the forces between slices.

3.1.1The factor of safety#

The factor of safety $F$ of a slip surface is the number by which the shear strength of the soil must be divided so that the strength that remains, the mobilised shear strength, is just enough to hold the sliding mass in equilibrium (Bishop, 1955). For a soil that follows the Mohr–Coulomb criterion in effective stresses, the shear stress mobilised at any point of the surface is

where

$\tau_f$

shear strength available on the surface (kPa)

$c',\ \phi'$

effective cohesion (kPa) and effective friction angle (°)

$\sigma'_n = \sigma_n - u$

effective normal stress on the surface: total normal stress minus pore pressure (kPa)

Two consequences of this definition are worth keeping in mind. First, $F$ applies to the strength parameters, $c'$ and $\tan\phi'$, not to the loads. Second, every method of slices assumes that the same $F$ is mobilised along the whole surface: the slope is treated as a rigid-plastic body that fails all at once. A value of $F = 1$ means limit equilibrium; a value below one means that the strength assumed cannot hold the mass on that surface.

Other strength models (Hoek–Brown, Barton–Bandis, undrained strength…) replace the bracket in Eq. (3.1.1) by their own envelope $\tau_f(\sigma'_n)$; how OGR Slip2D linearises a curved envelope inside each method is described in Section 3.5.

3.1.2The method of slices#

The stresses along a slip surface depend on the weight of the ground above each point, which changes along the surface. The method of slices handles this by dividing the sliding mass into vertical slices, narrow enough for the base of each one to be treated as straight and its strength as uniform (Fellenius, 1927). Figure 3.1.1 shows the critical surface of a Spencer analysis divided into the 30 slices OGR Slip2D used to compute it.

The forces acting on one slice are drawn in Figure 3.1.2. Each slice carries its weight $W$, any external load on its top, the forces on its base and the forces exerted on it by the two neighbouring slices.

The base of a slice of width $b$ inclined at $\alpha$ has length $l = b\sec\alpha$. Writing the shear strength on that base in terms of forces, and dividing it by $F$, gives the mobilised shear force that appears in every method:

The normal force $N$ is unknown: it depends on how much of the weight is carried by the base and how much by the neighbouring slices, and that is where the methods differ.

3.1.3Equations and unknowns#

For a mass divided into $n$ slices, statics provides three equations per slice (horizontal force, vertical force and moment equilibrium), so $3n$ equations in all. The unknowns are counted in Table 3.1.1, following Fredlund & Krahn (1977) and Duncan, Wright & Brandon (2014).

The shear force on each base is not an extra unknown, because Eq. (3.1.2) ties it to $N$ and $F$. Even so, there are $4n - 2$ unknowns for $3n$ equations: the problem has $n - 2$ more unknowns than equations and is statically indeterminate. Every method of slices closes it with an assumption about the interslice forces (their size, their inclination or both), and that assumption decides which equilibrium equations the method can still satisfy.

3.1.4How the methods close the problem#

Table 3.1.2 summarises the nine methods implemented in OGR Slip2D. Methods that satisfy only one kind of equilibrium are simpler and, for most slopes, close to the rigorous ones; methods that satisfy both force and moment equilibrium need an iteration on two quantities but make the smallest assumption. Each method is developed in Section 3.2.

3.1.5Direction of sliding#

The equations are written for a mass that slides in a known direction. The sign of the base inclination $\alpha$, the sense of the mobilised shear and the side on which a tension crack or a support acts all follow from it. OGR Slip2D therefore asks for the failure direction of the model (left to right when the ground falls to the right) and warns when the setting contradicts the geometry, for instance when the crest it implies is lower than the toe.

3.1.6What a factor of safety does not tell#

• No deformations. Limit equilibrium says whether the mass can stand; how much it moves before it fails has to be found with a deformation analysis or, under earthquake loading, with a sliding-block estimate (Section 3.9).
• A single, uniform $F$. Strain-softening materials and progressive failure, where different parts of the surface reach their peak strength at different times, are outside the method.
• Plane strain. A two-dimensional section ignores the resistance of the ends of a three-dimensional slide; for most slopes this is conservative.
• Only the surfaces tried. The critical factor of safety is the lowest one found. How OGR Slip2D searches for it, and how to check that the search has found it, is the subject of Section 3.4.

3.1.7References#

• Bishop, A.W. (1955). The use of the slip circle in the stability analysis of slopes. Géotechnique, 5(1), 7–17.
• Duncan, J.M., Wright, S.G. & Brandon, T.L. (2014). Soil Strength and Slope Stability, 2nd ed. Wiley.
• Fellenius, W. (1927). Erdstatische Berechnungen mit Reibung und Kohäsion (Adhäsion) und unter Annahme kreiszylindrischer Gleitflächen. Ernst & Sohn, Berlin.
• Fredlund, D.G. & Krahn, J. (1977). Comparison of slope stability methods of analysis. Canadian Geotechnical Journal, 14(3), 429–439.