Slope Stability: Example Analysis

Slope Stability Analysis

In this chapter of the Geoengineer.org series on Slope Stability, a simple example of slope stability analysis will be illustrated. The analysis will be conducted in Slide2 v6.0, a commercial 2D slope stability software by Rocscience that utilizes the Limit Equilibrium Method (LEM) and can be used for the analyses of soil and rock slopes, dams, embankments and retaining walls. Since the analysis will be two-dimensional, a representative cross-section of an idealized slope is developed to derive meaningful insights.

More specifically, a 1:1 slope (45 degrees inclination) that can be either natural or man-made will be considered. The slope height and width will both be 10 meters and the ground conditions will be comprised by two soil layers, a silty clay and a clay. The shear strength of both soils will be assessed via the Mohr-Coulomb failure criterion, as:

Where t is the shear strength, σ_n is the effective stress, σ_t is the total stress, u is the pore-water pressure, c is the cohesion and φ is the friction angle. In actual real-world cases, the strength parameters of the soil can be determined in drained, triaxial or direct shear tests. The analysis will be conducted assuming long-term conditions; therefore, the effective stresses will be used. In case of short-term analysis, and if undrained conditions are presumed, the undrained shear strength of the ground c_u is used (c=c_u and φ=0°) in the LEM analysis.

The soil parameters that are assumed for the slope stability analysis are given in Table 1.

Table 1. The soil parameters used in the example analysis

The boundaries of the model are placed at a distance of 15 meters from the toe and the crown of the slope, respectively, so that the critical failure surface search will not be affected. The initial geometry of the ground, as inserted in Slide2, is illustrated in Figure 1.

The objective of a slope stability analysis is to detect the critical failure surface and derive its Factor of Safety (FoS). If the FoS is greater than the minimum acceptable FoS, as defined by the codes/standards/regulations and the significance of the project, the slope is considered stable and no further actions may take place. On the contrary, if the FoS is found to be inadequate, support measures need to be implemented. For the purposes of the present simplified example, an indicative FoS greater than 1.25 will be required for the slope to be considered stable.

Slide2 enables the utilization of multiple limit equilibrium methods (e.g., Ordinary, Bishop, Janbu, Spencer, Morgenstern-Price). Here, the Bishop and the Spencer methods will be used, for illustration purposes. The number of slices for each surface analyzed is set to 30 and the maximum number of iterations to 50.

The groundwater table level may significantly vary based on the time of the year and other environmental conditions. In this example, 2 groundwater table cases will be assumed:

1. A favorable condition, in which groundwater table is relatively low and does not affect the failure surfaces,  and
2. An unfavorable condition, in which most of the ground lies below the groundwater table (Figures 2a and 2b).

In the second case, saturated conditions lead to pore-water pressure build-up, a fact that reduces the shear strength of the ground and consequently the FoS of the slope. For simplification reasons, the first conditions will be referred to as “Dry conditions” and the second as “Partially saturated conditions”.

At this stage, if any additional external load impacts the stability of the slope, it should be added in the analysis. In our case, we do not consider any external loads. Finally, an auto-generated or a manual grid based on which Slide2 will derive the failure surfaces, is determined. The grid should be large enough to ensure that the critical surface will be included in it.

The problem is currently well-defined and the computational process may begin. The FoS of the slope in dry and partially saturated conditions is derived via the Bishop and Spencer methods and the findings on the critical search surface are given in Figures 3a to 3d. The factors of safety for each case are given in Table 2.

Table 2. Factors of safety for the example slope, given the water table conditions and the limit equilibrium methods considered.

Based on the results of the Limit Equilibrium analyses, we derive the following conclusions:

1. In all cases (dry and partially saturated conditions), the critical surface is the same. It initiates about 3 meters behind the slope’s crown and ends up at its toe.
2. The factors of safety computed by either the Bishop or the Spencer method are almost identical,  given the groundwater conditions (dry or partially saturated).
3. In all cases, the failure surface propagates through both soil layers. Its depth is significant and cannot be neglected (there are cases that shallow failure surfaces are not taken into consideration due their minor impact on the engineering project).
4. Given that the limit value of the FoS in the example project is indicatively set to 1.25, the slope is considered marginally stable in dry conditions and unstable in partially saturated conditions, hence support measures should be implemented.

Slope Stabilization

In practice, an engineer must always ensure the stability of a slope given the worst-case scenario. In our simplified example, the slope is unstable in partially saturated conditions; hence, support measures must be applied so that its FoS exceeds the indicative limit value (1.25).

Many techniques to stabilize a slope and increase its FoS have been developed and employed over the years. Some of the most notable of them include the installation of drainage features, the construction of retaining walls or buttresses, the utilization of geotextiles or anchors, etc.

An optimal design also accounts for the financial feasibility of an engineering project thus, the objective of applying support measures is to increase the FoS with the minimum required and/or available resources. Hence, the selection of the most appropriate support measures is a case-specific issue.

Here, a ground stabilization technique known as soil nailing will be used to increase the FoS of the example slope. This method involves drilling holes for steel bars to be inserted into the face of the slope. The holes are subsequently grouted. As passive supports elements, soil nails are frequently utilized to stabilize slopes in soils and weak rocks. The technical characteristics that need to be defined for each soil nail are the following:

• Length: Soil nails usually measure 0.8 to 1.2 of the slope’s height, however, practically, they should not exceed 15 meters due to the difficulty in forming and maintaining the stability of the drillhole (Bridges, 2017).
• Tensile Capacity: The maximum tensile load that the soil nail can withstand, a values that depends on the material’s properties (steel) and its cross-sectional area.
• Plate Capacity: The maximum load that the plate which connects the soil nail and the ground can bear.
• Shear and Compression Capacities (optional): The maximum shear and compression loads that the soil nail can withstand without failing.
• Pullout Strength: The pullout stripping force which can be produced by a soil nail.

Regarding the geometry of the grid of soil nails that will be established to support a slope, additional parameters need to be determined. These are:

• Out-of-plane spacing: It defines how many meters will separate the soil nails in the 3^rd dimension (perpendicular to the 2D cross-sectional view).
• Longitudinal spacing: The longitudinal distance between soil nails measured i) along the boundary, ii) horizontally or iii) vertically.
• Orientation of the soil nails: The angle of the soil nails usually measured from the horizontal direction.
• Spatial distribution of the soil nails: At which points on the slope the support measures begin and end.

The assumed values of the aforementioned characteristics used for the simplified example analysis presented herein are shown in Table 3.

Table 3. Soil nail characteristics used in the example analysis

The slope supported by the soil nails is illustrated in Figure 4. Four soils nails are assumed in the 2D cross-sectional area per meter (in the third dimension). Hence if the width of the slope is 10 meters, the support measures would require 40 soil nails. The first soil nail is installed 1.7 meters measured from the crown of the slope and the last one 3.4 meters from its toe (Figure 4).

The limit equilibrium analysis, as described above, is repeated to derive the new FoS of the slope. The analysis yields a different critical failure surface and the new FoS is slightly above the assumed required value (1.25) as shown in Figures 5a and 5b. The new factors of safety for each case are given in Table 4.

Table 4. New factors of safety for the slope, after the support measures are applied.

The results of the analyses yield the following findings:

1. The new FoS considering both the Bishop and the Spencer method is above the limit value (1.25), hence, the support measures are considered acceptable.
2. Both methods yield the same critical failure surface.
3. The critical failure surface does not intersect with the soil nails and thus, it is longer and deeper than the one found by assuming an un-reinforced slope (Figure 3).
4. The installation of the soil nails increased the FoS of the slope by 59% and 61% considering the utilization of the Bishop and Spencer method, respectively.

To complete the assessment of the applied support measures, the new FoS in dry conditions is also derived. It can be reasonably assumed that since the measures are adequate in partially saturated conditions, the slope will also be stable in dry conditions. Nonetheless, it is useful to derive its FoS for comparison reasons. The results are given in Figure 6a and 6b and in Table 4.

When the support measures are applied, the factors of safety in dry conditions increase by 59% and 60% considering the Bishop and Spencer method, respectively. 

The support measures are acceptable in both groundwater conditions considered in this example analysis and could be implemented to stabilize the slope.

We emphasize that all the parameters given in this example (shear strength characteristics of the ground, tensile and shear strength of the soil nails, plate capacity etc.) refer to their design values after safety factors have been applied in accordance with the relevant design standards/codes/regulations.

References

Alsubal, S., Harahap, I.S.H.  and Babangida, N.M. (2017). A Typical Design of Soil Nailing System for Stabilizing a Soil Slope: Case Study. Volume: 10, Issue: 4, Pages: 1-7. DOI: 10.17485/ijst/2017/v10i4/110891

Bridges, C. (2017). Design of Soil Nailed Walls According to AS4678-2002. 8th Australian Small Bridges Conference

Tutorial: Rocscience/Slide

Tutorial: Rocscience/Soil Nails