Introduction
Soil slopes have naturally occurred due to geological procedures taking place for millions of years. The creation of a slope on the soil is the result of internal friction and cohesion of the soil in which particles resist the tendency to move due to gravitational forces. Due to human activity, slopes can be the result of intentional and non-intentional engineering interventions. Intentional engineering interventions can be excavations, landfills, cuttings, open-pit mining, embankments, etc. Non-intentional actions are typically related to poor design which leads to increased surcharge on the soil, exceedance of the soil’s shear strength and results in mass movements, finding a new state of equilibrium. Shear strength may also be decreased by the effects of weathering or changes in pore water pressure, which occur either naturally or due to human activity.
In a slope stability analysis, an engineer investigates the potential failure mechanisms and the slope’s sensitivity to various triggering mechanisms. Slope stability analysis involves the design of slopes considering safety, reliability and economics and the design of possible remedial measures. Geological information such as soil or rock properties and groundwater conditions, as well as site characteristics (e.g. slope geometry) need to be considered in the design of slopes. Especially the groundwater conditions need to be well known, as the presence of water can have an extremely detrimental effect on the soil properties and, so on the slope stability.
Slope Failure
Slope failure occurs when the maximum capacity of the soil is exceeded. The failure surface typically has a circular or ellipsoid shape for uniform soil properties, or non-circular shape if a weaker or stronger plane exists within the soil mass. The weaker soil layer fails, causing the movement of the mass above it.
Depending on the characteristics of the slope failure surface, four types can be defined:
- movement of soil mass along a thin layer of weak soil,
- base slide,
- toe slide, and
- slope slide.
Fig. 2.1 shows the four types of slope failure:
Figure 2.1. Types of slope failure: (a) movement of soil mass along a thin layer of weak soil, (b) base slide, (c) toe slide, (d) slope slide.
The effects of slope failure can be limited to the natural environment with changes on the surface of the slopes (Fig. 2.2 to 2.4), or affect the built environment with damage to retaining structures (Fig. 2.5), buildings, and infrastructure (Fig. 2.6 & 2.7) or even cause injuries and casualties when the affected region is in use.
Landslide at an embankment
Landslide on Mars
Wikimedia / Public Domain
Factors of Safety
The factor of safety is typically defined as the ratio of the maximum load or stress that a material (in this case the soil) can sustain to the actual load or stress that is applied on it. Fig. 2.8 shows Mohr’s circle for a typical cohesive soil. If τ is the shear stress on the soils and τff the shear stress that it could receive for the same amount of normal stress, the factor of safety can be defined as:
Figure 2.8. Mohr’s circle for a typical cohesive soil (Papavasileiou, 2020) .
Used under Fair Dealing.
If σn is the normal stress on the soil, substituting the maximum shear stress with the respective normal stress, the above formula is transformed as follows:
Two other factors of safety which are used occasionally are:
- the factor of safety with respect to cohesion ( Fc ), and
- the factor of safety with respect to friction ( Ff ).
The factor of safety with respect to cohesion ( Fc ) is defined as the ratio of the actual cohesion to the cohesion required for stability, when the frictional component of strength is fully used. Using this factor of safety in the above formula, it is:
The factor of safety with respect to friction ( Fc ) is the ratio of the tangent of the angle of shearing resistance of the soil to the tangent of the actual angle of shearing resistance of the soil when the cohesive component of strength is fully used. Considering this safety factor, the formula for the shear stress becomes:
Another factor of safety sometimes used is the factor of safety with respect to height (FH). This is the ratio between the maximum height of a slope to the actual height of a slope and may be expressed as follows:
From the aforementioned four factors of safety, the one more often used is the factor of safety with respect to the soil’s strength. The other three are rarely used in slop stability analysis.
Culmann Method
Culmann proposed a method for the analysis of the stability of slopes, which assumes a plane surface of failure through the toe of the slope (Fig. 2.9). This method of slope stability analysis is rarely used, because virtually plane surfaces of sliding are observed only on very steep slopes. For slopes with relatively small angles with the horizontal, the surfaces of sliding are almost always curved.
Figure 2.9. Culmann’s approach to slope stability .
In Fig. 2.9, line QS represents a potential plane failure surface. The wedge QRS is in equilibrium under the forces applying on it, namely:
- the weight of the wedge W,
- the developed cohesive force Cm, and
- the developed frictional force P.
The equilibrium of the aforementioned can be expressed as:
The ratio of cm/ρgH is known as the stability number. The plane QS at an angle θ to the horizontal is arbitrarily selected. So, the plane at which the resistance of the wedge to sliding is minimised is the failure plane. Setting the derivative of the above function to zero, the critical inclination (θcrit) is calculated as:
Substituting θcrit into the initial equilibrium equation yields the maximum value for the stability number:
Using a trial and error process, it is possible to define the safety factor with respect to strength.
Analysis Approach for Rotational Failure Surface
As discussed previously, most failure surfaces have been found to be curved. It was noticed that in a cross-section, the majority of them can be simulated by an arc of a circle. Various analytical methods have been proposed, which consider this shape and calculate the equilibrium of moments about the fictitious centre of this circle (Fig. 2.10). The forces which tend to cause the sliding of the wedge are the weight of the wedge and the resultant of the normal stresses on the failure surface. The force which resists siding is the resultant of the shear stresses that act on the failure surface. The factor of safety can be calculated as the sum of stabilising moments, over the sum of destabilising moments on the wedge, or:
It can be noticed that the normal stresses are not taken into account in the formula above. This is because the normal forces act perpendicular to the failure surface, which means that their direction passes from the centre of the circle. Hence, their moment is zero.
Figure 2.10. Illustration of the forces/stresses considered in the assessment of stability of slopes with curved failure surface.
Also, it should be noted that the maximum shear stress is considered to be a constant, i.e. it does not variate along the failure surface. If there is variable maximum shear strength, then the integral needs to be calculated along the failure surface:
The φ = 0 method
If the soil is saturated clay, the angle of short-term shearing resistance (φu) is zero. Hence, the maximum resisting shear stress around the failure surface will be equal to the undrained cohesion (cu). If the undrained cohesion is non-variable around the failure surface, then the factor of safety can be calculated as:
This method has been widely used in practice to assess the short-term stability of saturated clay slopes. Due to the assumption that φu = 0, this method is referred to as ‘The φ = 0 method’.
Figure 2.11 . Illustration of the forces considered in the assessment of stability of slopes with curved failure surface.
If the angle of shearing resistance is not equal to zero, then the frictional component of the resisting shear stresses needs to be calculated. As illustrated in Fig. 2.11, the forces acting on the wedge, are:
- the developed cohesion force (C'm),
- the effective normal force (N'),
- Rφ which is the frictional force acting around the arc (= N'×tanφ'/F)
However, the safety factor F, the magnitude and the line of action of N’ and Rφ are unknown. The magnitude of Rφ can be calculated as:
Rφ = N'×tanφ'/F
So, there are four unknown parameters in total. Since there are only three equations of static equilibrium, the problem is indeterminate. To solve the problem, typically an assumption on the distribution of the effective normal stress is made, so that the direction of N’ is determined.
Ordinary Method of Slices
When the effective angle of shearing resistance is not constant along the failure surface, the previous method cannot be used. It was discussed that, to consider for the variability, of soil properties, one would need to integrate. A close approximation to integration often used in mathematics is the trapezoid rule: the surface is divided into an adequate number of trapezoids, the parallel sides of which are in the vertical direction, so that their combination resembles closely the actual surface. The larger the number of trapezoids considered, the better the approximation to the actual solution.
The same principle is used in the case of slope stability analysis. This method is called ‘The method of slices’. The block that is above the failure surface and would slide at failure, is divided into a number of trapezoids (slices), so that the normal and shear stresses at the failure surface can be calculated. The curved bottom of each slice is approximated as chord. The bottom of each slice passes through one type of material, so its mechanical properties remain constant. The larger the number of slices, the better the approximation can be. The factor of safety is calculated in the conditions that minimize the resisting forces (i.e. the stabilization moments). Figure 2.12 shows the forces acting on a single slice.
Figure 2.12. Illustration of the forces considered in the assessment of stability of slopes with curved failure surface.
Because the slice is in equilibrium, it is assumed that the side force pairs (En - En+1 and Xn - Xn+1) are equal in magnitude, so they can be neglected in the calculations, as they cancel each other out. It is also assumed that the normal force N may be determined by resolving the weight W of the slice in a direction normal to the arc, at the midpoint of the slice, as shown in Fig. 2.12. So:
where a is the angle of inclination of the potential failure arc to the horizontal at the midpoint of the slice.
The effective normal force is:
The total maximum resisting force is:
So, the factor of safety is:
The above calculations are performed a number of time for different trial failure surfaces until the lowest factor of safety is found.
The following table can be used to calculate the sum of forces at each step of the procedure:
Bishop Method of Slices
Bishop (1955) proposed a variation of the slices method which gives different answers than the ‘Ordinary Method of Slices’. In Bishop’s method, stresses are used instead of forces. The main difference between ‘Bishop’s Method’ and the ‘Ordinary Method of Slices’ is that the resolution of forces takes place in the vertical direction instead of a direction normal to the arc.
The procedure is as follows:
To find σ' resolve forces in the vertical direction:
Where the factor of safety is:
where
As the factor of safety appears on both sides of the equation, a fast-converging iterative procedure is followed.
Bishop’s Method yields factors of safety which are higher than those obtained with the Ordinary Method of Slices. Also, the two methods do not lead to the same critical circle. It has also been found that the disagreement increases as the central angle of the critical circle increases. Analyses by more refined methods involving consideration of the forces acting on the sides of slices show that the simplified Bishop Method yields answers for factors of safety which are very close to the correct answer.