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16-Civ-A4 Geotechnical Materials and Analysis · December 2014

Question 2 of 6: The three limiting lateral earth pressures

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format: National Examinations (Engineers Canada / PEO), 98-Civ-A4 Geotechnical Materials and Analysis, December 2014. Closed book, 3 hours, 100 marks. Six questions — answer all. Charts and equations supplied at the back of the paper.

Reference texts: Das & Sobhan, Principles of Geotechnical Engineering (9th ed.), Cengage; Holtz, Kovacs & Sheahan, An Introduction to Geotechnical Engineering (2nd ed.), Pearson; Craig’s Soil Mechanics (Knappett & Craig, 8th ed.), CRC Press.

Check: The label inside Figure 3 prints “$k = 2.0 \times 10^{5}$ m/s”, which is physically impossible (200 km/s). The text of Question 5(b) prints $k = 2.0 \times 10^{-5}$ m/s, so the figure label has lost its minus sign and $2.0 \times 10^{-5}$ m/s is used throughout Question 5.

Question 2: The three limiting lateral earth pressures (10 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.

Wall movement and the three limiting states— no strain —At-rest (K₀)no movementActive (Kₐ)wall moves away from soilPassive (Kₚ)wall pushed into soil
Figure Q2a. At-rest requires no wall strain; the wall must yield away from the backfill to reach the active state and be driven into it to mobilise the passive state.

The horizontal effective stress on a wall is not a fixed quantity — it depends on how much the wall has moved. Starting from the undisturbed “at-rest” condition, only a tiny strain is needed to reach the active limit, whereas a much larger inward movement is required to mobilise the passive limit. The three states are bracketed by the earth-pressure coefficient $K = \sigma'_h/\sigma'_v$.

At-rest ($K_0$). The wall does not yield, so the soil is laterally confined exactly as it was in the ground. No shear failure is mobilised, $\sigma'_h = K_0\,\sigma'_v$ with $K_0 = 1-\sin\phi'$ for a normally consolidated soil (larger for over-consolidated soil). Because both principal stresses are set by the confinement, the Mohr circle is small and sits comfortably inside the failure envelope.

Active ($K_a$). If the wall yields away from the backfill, the soil expands laterally, $\sigma'_h$ falls while the vertical stress $\sigma'_v$ (the major principal stress) is unchanged. The Mohr circle grows until it just touches the failure envelope: failure by lateral extension. For a cohesionless soil $\sigma'_h = K_a\,\sigma'_v$ with $K_a = \tan^{2}\!\left(45^\circ - \tfrac{\phi'}{2}\right)$. This is the minimum possible lateral pressure and it is the value used to design the wall stem.

Passive ($K_p$). If the wall is driven into the soil, $\sigma'_h$ rises until it exceeds $\sigma'_v$; the horizontal stress becomes the major principal stress and the soil fails by lateral compression. Now $\sigma'_h = K_p\,\sigma'_v$ with $K_p = \tan^{2}\!\left(45^\circ + \tfrac{\phi'}{2}\right) = 1/K_a$. This is the maximum lateral resistance, mobilised in front of embedded walls and by passive keys.

Mohr circles: at-rest, active and passiveσ′ (kPa)τfailure envelope τ = σ′ tanφ′σ′ᵥσ′ₐK₀σ′ᵥσ′ₚAt-restActivePassive
Figure Q2b. The at-rest circle sits well inside the envelope. Reducing the horizontal stress swings the active circle onto the envelope (σ′ᵥ is the major stress); increasing it drives the passive circle onto the envelope (σ′ᵥ becomes the minor stress).

On the Mohr diagram the vertical stress $\sigma'_v$ is common to all three circles. The at-rest circle lies inside the envelope; swinging $\sigma'_h$ down to $K_a\sigma'_v$ brings the circle onto the envelope with $\sigma'_v$ as the major stress (active), and pushing $\sigma'_h$ up to $K_p\sigma'_v$ brings a much larger circle onto the envelope with $\sigma'_v$ now the minor stress (passive). Because $K_p = 1/K_a$, the passive resistance is many times the active pressure — for $\phi' = 30^\circ$, $K_a = \tfrac13$ and $K_p = 3$, a nine-fold range.