Slope Stability

Slope stability is a paramount concern in geotechnical engineering, particularly for natural hillsides, man-made embankments (like dams, levees, and highway fills), and deep excavations. Failure occurs when the shear stresses driving the soil downward (due to gravity, seepage forces, or external loads like earthquakes) exceed the shear strength resisting the movement along a potential slip surface. Understanding these driving and resisting forces allows engineers to assess the safety of slopes and design appropriate mitigation measures.

Slope failures can be broadly categorized based on the geometry of the slip surface and the type of material.

If the calculated factor of safety is inadequate (typically $FS < 1.5$), mitigation measures are necessary to either decrease the driving forces or increase the resisting forces.

An infinite slope represents an idealization where the slope extends uniformly over a considerable distance. The soil properties and any groundwater conditions are assumed constant at any given depth parallel to the slope surface. Consequently, the critical failure surface is assumed to be a plane parallel to the slope face at some depth $z$.

For a simple dry sand slope inclined at angle $\beta$, the factor of safety against sliding along a plane at depth $z$ is independent of depth.

Simplifying the previous expression yields the fundamental equation for an infinite cohesionless slope.

If groundwater is seeping parallel to the slope face and emerges at the surface (the worst-case scenario for infinite slopes), the pore water pressure significantly reduces the effective normal stress ($\sigma'$), and consequently, the shear strength.

For finite slopes (e.g., dams, embankments, cuts), the critical slip surface is rarely a straight plane. It is typically curved, often assumed to be an arc of a circle for analytical simplicity, especially in cohesive soils.

For saturated clays under undrained conditions, the internal friction angle is assumed to be zero ($\phi = 0$). In this case, the shear strength is simply the undrained shear strength, $s_u$ (or $c_u$). The slip surface is assumed to be a circular arc of radius $R$. The resisting moment is provided by cohesion along the entire arc length $L$, and the driving moment is from the weight of the sliding mass $W$ acting at a horizontal distance $x$ from the center of rotation.

The most common approach for analyzing complex finite slopes (with $c'$ and $\phi'$) is the Method of Slices (e.g., Fellenius/Ordinary method, Bishop's simplified method, Spencer's method). The sliding mass above a trial circular slip surface is divided into numerous vertical slices.

The basic principle involves formulating equilibrium equations (force and/or moment) for each slice and then integrating over the entire sliding mass.

The Ordinary Method of Slices, developed by Fellenius, assumes that the resultant of the inter-slice forces (forces between adjacent vertical slices) is completely parallel to the base of each slice, effectively ignoring them for the calculation of the normal force on the base.

This method often underestimates the factor of safety (is overly conservative) for deep circles or high pore pressures.

The Ordinary Method of Slices (Fellenius method) ignores the inter-slice forces (forces between adjacent slices), which can lead to conservative (lower) estimates of the Factor of Safety. Bishop's Simplified Method assumes that the inter-slice shear forces are zero but accounts for inter-slice normal forces. This provides a more accurate and widely accepted result.

While Bishop's Simplified Method is excellent for circular slip surfaces, it does not satisfy horizontal force equilibrium. Methods like Spencer's Method and the Morgenstern-Price Method are more rigorous. They satisfy all conditions of equilibrium (forces and moments) and can be used for any shape of slip surface (circular or non-circular). These methods calculate an inter-slice force function, yielding highly accurate FS values, but require specialized software to solve the complex iterative equations.

Earthquakes impart horizontal and vertical inertial forces on slopes, severely reducing stability. A pseudo-static analysis simulates this by applying a constant horizontal force ($F_h = k_h \cdot W$) and vertical force ($F_v = k_v \cdot W$) to the center of gravity of the sliding mass. The horizontal seismic coefficient ($k_h$) is typically a fraction of the peak ground acceleration (PGA).

Any single limit equilibrium calculation only yields the FS for one specific trial circle. The true Factor of Safety of the slope is the minimum FS possible. Therefore, engineers must use search algorithms (like grid search for circular centers, auto-refine, or simulated annealing for non-circular shapes) to analyze thousands of potential slip surfaces to locate the critical slip surface.

Rapid drawdown occurs when the water level against a slope (e.g., inside a reservoir behind an earth dam) drops very quickly. The water pressure supporting the face of the slope is removed instantly, but the pore water pressure inside the soil mass remains high because it takes time to drain out. This excess pore water pressure drastically reduces the effective stress and shear strength, making rapid drawdown one of the most critical design conditions for earth dams.

For simple, homogeneous finite slopes, D.W. Taylor developed a dimensionless Stability Number ($N_s$) based on the friction circle method. This allows for quick, chart-based determination of the Factor of Safety without needing to slice the slope. By looking up $N_s$ in Taylor's charts based on the slope geometry and soil friction, the critical height or the Factor of Safety for a given height can be rapidly determined.

In purely cohesive soils (clays) or c-$\phi$ soils, active earth pressures can become negative near the ground surface. This leads to the formation of vertical tension cracks extending downwards from the top of the slope. These cracks significantly reduce the overall stability because they truncate the potential slip surface (reducing the area over which cohesion can act to resist sliding) and can fill with rainwater, adding immense hydrostatic driving pressure pushing the sliding mass outward.

For a more rigorous analysis of slip circles, especially in non-homogeneous soils or varying groundwater conditions, the sliding mass is divided into vertical slices.

• Ordinary Method of Slices (Fellenius Method): The simplest method. It assumes that the interslice forces (the forces between adjacent vertical slices) are purely horizontal and equal and opposite (meaning they cancel out), which is physically inaccurate but computationally easy. It tends to provide a conservative (lower) Factor of Safety.
• Bishop's Simplified Method: A significantly more accurate method. It satisfies vertical force equilibrium for each slice and overall moment equilibrium about the center of the slip circle. It assumes the interslice shear forces are zero but accounts for interslice normal forces. It is the most widely used method in standard geotechnical practice for circular slip surfaces.