Question 3 of 6: Lateral Earth Pressures / Slope Stability

Question 3: Lateral Earth Pressures / Slope Stability (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) At-rest, active and passive earth pressure coefficients

A retaining structure interacts with the soil behind it through a lateral earth pressure coefficient $K$, defined as the ratio of horizontal to vertical effective stress, $K=\sigma_h'/\sigma_v'$. The value of $K$ depends entirely on how much lateral strain the wall allows the soil to undergo.

At-rest ($K_0$). The wall does not move at all — the soil is in its natural, undisturbed state with zero lateral strain. Jaky's empirical relation gives $K_0\approx1-\sin\phi'$ for normally consolidated soil. Example: a rigid basement wall braced top and bottom before any backfill settlement occurs, or a soil element far from any excavation.

Active ($K_a$). The wall moves away from the retained soil by a small amount, allowing the soil to expand laterally and mobilize its full shear strength in the direction that reduces the horizontal stress. Rankine's relation gives $K_a=\tan^2(45^{\circ}-\phi'/2)$, the smallest of the three coefficients. Example: a free-standing cantilever retaining wall that rotates slightly forward under backfill pressure, or a basement wall once the building settles enough to relieve some pressure.

Passive ($K_p$). The wall is pushed into the soil, compressing it laterally until the soil's shear strength is mobilized in the opposite sense, resisting the movement. Rankine's relation gives $K_p=\tan^2(45^{\circ}+\phi'/2)$, the largest of the three coefficients ($K_p=1/K_a$ for the same $\phi'$). Example: the soil in front of a sheet-pile wall's embedded toe, which must be pushed aside as the wall kicks outward, or the resistance mobilized in front of a bridge abutment under thermal expansion.

Since $K_a

(b) General approach of limit equilibrium slope stability methods

Every limit equilibrium method shares the same underlying logic, regardless of which specific method (Ordinary/Fellenius, Bishop's Simplified, Janbu, Spencer, Morgenstern–Price) is used:

1. Assume a trial failure surface. A potential slip surface (commonly circular for homogeneous soil, or a general non-circular surface for layered ground or a known weak plane) is assumed to divide the slope into a moving mass above and a stable mass below.
2. Discretize into vertical slices. The moving soil mass above the trial surface is divided into a series of vertical slices, each with its own weight, base inclination, base length and pore pressure.
3. Compute driving and resisting forces. The weight of each slice generates a driving (destabilizing) shear force along the base of the slice; the soil's Mohr–Coulomb shear strength ($\tau_{\text{rupt}}=c'+\sigma'\tan\phi'$) mobilized along that same base generates the resisting force.
4. Enforce equilibrium. Force and/or moment equilibrium is written either for the slope mass as a whole or slice-by-slice, with different methods making different simplifying assumptions about the interslice forces between adjacent slices (Fellenius ignores them entirely; Bishop assumes horizontal interslice forces with zero net vertical shear; Spencer and Morgenstern–Price satisfy full equilibrium by assuming an interslice force function).
5. Compute the factor of safety. $$FS=\frac{\text{sum of resisting (shear strength) forces or moments along the surface}}{\text{sum of driving (gravity-induced) forces or moments along the surface}}$$
6. Search over trial surfaces. Steps 1–5 are repeated for many trial surfaces (varying centre, radius, or geometry), and the surface giving the minimum $FS$ is taken as the critical (most likely) failure surface.

The methods differ mainly in how many equilibrium equations they satisfy and what they assume about the interslice forces — more rigorous methods (Spencer, Morgenstern–Price) satisfy both force and moment equilibrium and are more accurate for non-circular or complex surfaces, while simpler methods (Fellenius, Bishop) are computationally lighter but can be less accurate, particularly for high pore pressures or steep, non-circular surfaces.