5.3 Slope Stability and Soil Stabilization

Fundamentals of Slope Stability

Evaluating slope stability is a critical component of geotechnical engineering, involving the analysis of natural or man-made slopes (such as highway cuts, railway embankments, and earth dams) to determine their resistance to gravitational sliding and shear failure. A slope fails when the driving stresses (primarily gravitational forces and water pressures) exceed the resisting forces (the shear strength of the soil along the potential failure surface).

The stability of a slope is quantified using the factor of safety (FS), defined as the ratio of the ultimate shear strength of the soil (tau_f) to the average shear stress (tau_d) developed along the failure surface:

FS = tau_f / tau_d

A slope is theoretically stable if FS > 1.0. However, in practice, engineers design slopes with an FS >= 1.3 for temporary slopes and FS >= 1.5 for permanent slopes.

Infinite Slope Analysis

An infinite slope represents a simplified model where the soil layers are uniform and parallel to the slope surface, and the boundary effects at the top and bottom of the slope are negligible. This model is highly relevant for analyzing shallow landslides and veneer failures.

Cohesionless Soils (c' = 0)

For dry or moist sand slopes with no water flow, the stabilizing force is purely frictional. Consider a soil block on a slope inclined at angle beta. The normal stress on the failure plane at depth H is sigma' = gamma * H * cos²(beta), and the shear stress is tau = gamma * H * sin(beta) * cos(beta). Applying the Mohr-Coulomb shear strength criterion (tau_f = sigma' * tan(phi')):

FS = (sigma' * tan(phi')) / tau = [gamma * H * cos²(beta) * tan(phi')] / [gamma * H * sin(beta) * cos(beta)] = tan(phi') / tan(beta)

where phi' is the angle of internal friction of the soil. This reveals that for dry cohesionless soils, the factor of safety is independent of depth H and soil unit weight. The slope is stable as long as the slope angle beta is less than or equal to the friction angle phi' (often called the angle of repose).

Cohesionless Soils with Seepage

When water flows through the slope, the presence of pore water pressure reduces the effective normal stress and adds a downslope seepage force. For steady seepage parallel to the slope face, with the groundwater table at the slope surface:

FS = [1 - (gamma_w / gamma_sat)] * [tan(phi') / tan(beta)] = (gamma' / gamma_sat) * [tan(phi') / tan(beta)]

where gamma' is the buoyant unit weight of soil, gamma_sat is the saturated unit weight, and gamma_w is the unit weight of water. Because the ratio gamma' / gamma_sat is typically around 0.5, seepage parallel to the slope face reduces the factor of safety of a cohesionless slope by approximately 50%.

Cohesive Soils (c' > 0, phi' > 0)

For soils containing both clay (cohesion c') and sand (friction phi'), the factor of safety at depth H is:

FS = c' / (gamma * H * sin(beta) * cos(beta)) + tan(phi') / tan(beta)

Here, the factor of safety depends on the depth H. As H increases, the cohesion term decreases, implying that failure will occur at a critical depth (H_cr) where FS = 1.0:

H_cr = c' / [gamma * (sin(beta)*cos(beta) - cos²(beta)*tan(phi'))]

Finite Slopes and Slope Failure Modes

Finite slopes have a limited height (such as a typical highway embankment or cut slope). The failure surface is typically curved rather than planar.

Common Slope Failure Modes

1. Rotational failure: The failure wedge rotates along a curved surface. This is common in homogeneous soils (especially clays).

   • Toe failure: The failure surface intersects the toe of the slope.
   • Slope failure: The surface intersects the face above the toe.
   • Base failure: The failure surface passes below the toe, common when a soft clay layer underlies the slope.

2. Translational failure: The soil mass slides along a planar failure surface. This occurs in stratified soils where a weak layer (such as silt or soft clay) exists parallel to the slope face or along rock bedding joints.

3. Flow failure: Occurs in loose, saturated sands or sensitive clays where liquefaction or high water pressure causes the soil to flow like a viscous liquid.

Analysis Methods: Method of Slices

For finite slopes with curved slip surfaces, the soil mass is divided into vertical slices to account for varying soil properties and stress conditions along the slip line. This approach is called the method of slices.

Ordinary Method of Slices (Fellenius Method)

This method assumes that the forces acting on the sides of any slice (interslice forces) balance each other out and can be neglected. While this simplifies calculations, it leads to conservative results. The factor of safety is:

FS = Sum([c' * delta_L + (W * cos(theta) - u * delta_L) * tan(phi')]) / Sum(W * sin(theta))

where:

• W is the weight of each slice.
• theta is the angle between the slice base and the horizontal.
• u is the pore water pressure at the base of the slice.
• delta_L is the length of the slice base.

Bishop's Simplified Method

Bishop's method assumes that the shear forces on the sides of the slices are zero, but accounts for the normal forces between slices. This method requires an iterative approach to solve for FS because the term FS appears on both sides of the equation, but it provides a much more accurate factor of safety than the Fellenius method:

FS = Sum([ (c' * b + (W - u * b) * tan(phi')) / (cos(theta) + sin(theta) * tan(phi') / FS) ]) / Sum(W * sin(theta))

where b is the width of each slice (b = delta_L * cos(theta)).

Soil Stabilization Techniques

When a slope is found to be unstable (FS < 1.3), geotechnical engineers implement soil stabilization techniques to increase the factor of safety.

1. Geometry Modification

Modifying the slope's geometry is the most straightforward stabilization method:

• Flattening the slope: Reducing the slope angle beta decreases the driving shear stress.
• Benching: Constructing intermediate horizontal steps (benches) along the slope face. Benches reduce the average slope angle, control surface water runoff, and catch falling debris.
• Toe berms: Adding a compact soil berm at the toe of the slope counteracts base failure by acting as a resisting surcharge block.

2. Drainage Systems

Because high pore water pressure is the most common trigger for landslides, lowering the groundwater table is highly effective:

• Horizontal drains: Perforated pipes drilled into the slope to drain perched water tables.
• Trench drains: Deep gravel-filled trenches along the top or face of the slope to intercept surface and subsurface water.

3. Structural Retaining Systems

Physically blocking soil movement using structural walls:

• Gravity walls or cantilever walls at the toe.
• Sheet pile walls or soldier pile walls driven deep past the failure surface to provide shear resistance.

4. Soil Nailing

Soil nailing is a technique where steel bars (nails) are installed into pre-drilled holes in the slope and grouted. A shotcrete facing is then applied to the slope surface. Unlike active anchors, soil nails are passive reinforcement elements; they only develop tension as the soil mass undergoes minor movement. This creates a reinforced gravity block that resists lateral sliding.

5. Geosynthetics and Geotextiles

Geosynthetics (including geotextiles and geogrids) are polymeric materials used to reinforce slopes, filter water, and separate soil layers:

• Geogrids: High-strength grids placed horizontally in compacted soil layers during embankment construction. The soil particles interlock with the grid openings, transferring tensile stresses to the geogrid and allowing the construction of very steep slopes.
• Geotextiles: Non-woven geotextiles act as filters, allowing water to pass through while keeping soil particles in place, which prevents internal erosion (piping). Woven geotextiles provide tensile reinforcement.