CELE Geotechnical Engineering — Lateral Earth Pressure and Retaining StructuresRevision Notes

Formulas

Exam Tips

• Memorize the trio: Kₐ = tan²(45 − φ/2), Kₚ = tan²(45 + φ/2), K₀ = 1 − sin φ. Verify Kₐ × Kₚ = 1 as a rapid self-check.
• For φ = 30°, the common values Kₐ = 1/3 and Kₚ = 3 appear very frequently in board problems — recognize them instantly.
• For φ = 0 (pure clay, short-term undrained): Kₐ = Kₚ = 1, meaning σₐ = σₚ = γz (pressure is isotropic in all directions before drainage).
• When Coulomb formulas appear in the exam, they are usually given — you need to substitute correctly, not derive from scratch.

Key Points

• Three pressure states exist depending on wall movement: at-rest (K₀), active (Kₐ), and passive (Kₚ).
• Active pressure is the SMALLEST lateral pressure — it develops when the wall moves AWAY from the backfill, allowing the soil to expand (shear failure in one direction).
• Passive pressure is the LARGEST lateral pressure — it develops when the wall is PUSHED INTO the soil, causing it to compress (shear failure in the opposite direction).
• At-rest pressure is intermediate — it acts on walls that cannot move (e.g., basement walls braced by floor slabs).
• Rankine theory assumes a smooth (frictionless) vertical wall and a horizontal backfill surface; it is the standard for most board exam problems.
• Coulomb theory is more general — it accounts for wall friction (δ) and sloping backfill (β), and is used when those conditions are specified.
• For a given φ, the relationship is always: Kₐ < K₀ < Kₚ.
• Note: Kₐ × Kₚ = 1, so Kₚ = 1/Kₐ (Rankine only).

Definitions

Section Title

Earth-Pressure States and Coefficients

Common Mistakes

• Using Kₐ on the passive (toe) side and Kₚ on the active (backfill) side — always match the correct coefficient to the correct side.
• Forgetting that Kₚ = 1/Kₐ (Rankine, cohesionless) — you can quickly verify your values with this reciprocal relationship.
• Applying Rankine formulas when the problem specifies wall friction δ ≠ 0 or a sloped backfill — in those cases, Coulomb's formula must be used.
• Confusing K₀ = 1 − sin φ (normally consolidated) with the overconsolidated case K₀ = (1 − sin φ)·OCR^(sin φ) — for standard board problems, use the simpler form unless told otherwise.

Formulas

Exam Tips

• Pressure diagrams: triangular for soil (acts at H/3), rectangular for surcharge (acts at H/2). DRAW the diagrams before computing — it prevents errors.
• When water is present, split the analysis: soil effective stress (γ' below WT) + hydrostatic water pressure (γw·hw). Add both force components to get total lateral thrust.
• For tension crack problems: find zc first, then compute the active thrust only over the zone from zc to H (the positive pressure zone).
• The H² dependence is a favorite exam trick — doubling H quadruples Pₐ. This is why tall walls are disproportionately more critical than short walls.

Key Points

• For a dry, cohesionless, homogeneous backfill of height H, the lateral pressure varies LINEARLY from zero at the top to Kₐγ H at the base — a triangular distribution.
• The resultant thrust Pₐ = ½ Kₐ γ H² acts at H/3 from the base of the wall (centroid of the triangle).
• A uniform surcharge q (kPa) on the backfill surface adds a RECTANGULAR pressure diagram of intensity Kₐ q over the full height H.
• The surcharge thrust ΔPₐ = Kₐ q H acts at H/2 from the base.
• Cohesion c reduces active pressure: σₐ = Kₐ γ z − 2c√Kₐ. The net pressure is negative near the top, indicating a tension zone. A tension crack forms to depth zc = 2c/(γ√Kₐ).
• Below the water table, use the submerged (buoyant) unit weight γ' = γsat − γw for soil stresses, PLUS add the full hydrostatic water pressure u = γw·hw separately.
• The total active thrust with water is the sum of: (1) active soil pressure using γ' below WT, (2) hydrostatic water pressure diagram (triangle below WT). These must be computed and added as separate resultants.
• Passive thrust: Pₚ = ½ Kₚ γ H² at H/3 from the base of the embedded depth.
• For layered soils, compute σᵥ incrementally through each layer and apply the appropriate K coefficient for that layer.

Definitions

Section Title

Resultant Thrust and Pressure Distributions

Common Mistakes

• Forgetting to add water pressure separately — a very common board exam trap. Always check if the problem says 'saturated backfill' or gives a water table location.
• Using γ (bulk unit weight) below the water table instead of γ' (buoyant unit weight) for the soil stress — this significantly overestimates soil pressure and underestimates water pressure.
• Computing the location of the resultant incorrectly when both a surcharge (acts at H/2) and soil (acts at H/3) are present — you must take moments about the base to find the combined resultant location.
• In cohesive soils, summing the full pressure diagram (including the tension zone) instead of only the positive pressure zone — the tension zone contributes no thrust.
• Doubling the height quadruples the thrust (H² relationship) — a 50% increase in wall height increases the thrust by 125%, not 50%.

Formulas

Exam Tips

• Organize your stability calculations in a TABLE: Component | Weight (kN/m) | Moment Arm from Toe (m) | Moment (kN·m/m). This prevents arithmetic errors under exam pressure.
• The three checks have different required FS: OT ≥ 1.5–2.0, slide ≥ 1.5, bearing ≤ qₐ. Remember the minimum FS for sliding is always 1.5.
• When passive resistance is given but you are conservative: FS(slide) without Pₚ < 1.5 but FS(slide) with Pₚ ≥ 1.5 — the board may ask 'is it safe?' Always clarify your assumption.
• If FS(OT) >> 2.0 and FS(slide) is borderline, consider adding a shear key to increase passive resistance — this is a common design recommendation asked in board exams.

Key Points

• Three external stability modes are checked for every retaining wall: overturning, sliding, and bearing capacity (eccentricity).
• All moments in the overturning and sliding checks are taken PER UNIT LENGTH of wall (kN·m/m and kN/m).
• The overturning moment acts about the TOE of the wall; all resisting moments are also taken about the toe.
• Minimum required factors of safety: FS(overturning) ≥ 1.5 to 2.0; FS(sliding) ≥ 1.5; bearing: resultant must fall within the MIDDLE THIRD of the base.
• Resisting forces in overturning include: weight of the wall stem, weight of the base footing, weight of soil above the heel, and any surcharge on the heel side.
• Active thrust provides the overturning moment (horizontal component × its moment arm from the toe).
• For sliding: resisting force = μ × ΣW, where μ = tan δ (base friction). A passive wedge at the toe adds resistance but is often NEGLECTED conservatively.
• For bearing: the resultant of all vertical loads must act within the middle third of the base width B. If it falls outside, tension develops in the base — generally not acceptable for unreinforced masonry or mass concrete walls.
• Middle third rule: eccentricity e = B/2 − x̄ ≤ B/6, where x̄ is the distance of the resultant from the toe.
• Toe and heel bearing pressures: q_toe = (ΣW/B)(1 + 6e/B), q_heel = (ΣW/B)(1 − 6e/B). The toe pressure must not exceed the allowable bearing capacity qₐ.

Definitions

Section Title

Retaining-Wall Stability Checks

Common Mistakes

• Taking overturning moments about the HEEL instead of the TOE — moments must always be about the toe for conventional stability checks.
• Forgetting to include the weight of soil ABOVE THE HEEL BASE SLAB in the resisting moment calculation — this soil contributes both vertical force and a moment arm about the toe.
• Including passive resistance at the toe without checking if it is reliable — if the problem says 'neglect passive resistance,' do not include Pₚ in FS(slide).
• Using the wrong moment arm for the active thrust — if Pₐ is inclined (Coulomb), separate it into horizontal (Pₐ,H) and vertical (Pₐ,V) components; the horizontal component causes overturning, and the vertical component may increase or decrease the resisting moment.
• Forgetting that eccentricity e is measured from the CENTER of the base, not from the toe — e = B/2 − x̄.

Exam Tips

• For φ = 30°: memorize Kₐ = 1/3, Kₚ = 3, K₀ = 0.5. These appear in at least 60% of lateral earth pressure board problems.
• For φ = 45°: Kₐ = tan²(22.5°) = 0.172, Kₚ = 1/0.172 = 5.83 — less common but has appeared in recent board exams.
• When two conditions are given (e.g., FS for overturning and sliding), compute both before concluding — a wall can pass one check and fail another.
• Time management: lateral earth pressure problems with full stability checks typically take 8–12 minutes. Budget accordingly for the 5-hour board exam.

Key Points

• PITFALL 1 — Wrong K coefficient: Kₐ drives the wall (backfill side); Kₚ resists at the toe (embedded portion). Never interchange them.
• PITFALL 2 — H² effect: Doubling the retained height multiplies the active thrust by 4, and the overturning moment by 8 (since Mₒ = Pₐ × H/3 ∝ H³). Extremely sensitive parameter.
• PITFALL 3 — Water: Always use γ' for soil below the water table AND add a separate hydrostatic pressure. Using γsat throughout is a common and costly error.
• PITFALL 4 — Location of resultant: Triangular soil pressure → H/3 from base. Rectangular surcharge → H/2 from base. Must use correct arm for moment calculations.
• PITFALL 5 — Cohesion: In active state, cohesion REDUCES pressure (subtract 2c√Kₐ). In passive state, cohesion INCREASES resistance. The tension crack depth is zc = 2c/(γ√Kₐ).
• STRATEGY: Always draw the pressure diagram first, then compute resultants, then sum moments about the toe in a table.
• SI units throughout: forces in kN, moments in kN·m, pressures in kPa, lengths in m. Never mix units.
• In Coulomb problems, the formula for Kₐ is given — substitute δ (wall friction) and β (backfill slope) carefully, watching sign conventions.

Section Title

Board-Exam Pitfalls and Quick Strategies

Common Mistakes

• Computing Pₐ = Kₐ γ H (forgetting the ½ factor) — the correct formula is Pₐ = ½ Kₐ γ H². This error underestimates the thrust by half.
• Using φ in radians instead of degrees when computing sin φ and tan φ — always use degrees for Rankine coefficients.
• Forgetting to convert surcharge from kN/m² to the correct units when computing ΔPₐ = Kₐ q H — the units should yield kN/m (force per unit length of wall).
• Not verifying the Kₐ × Kₚ = 1 identity after computing both — if it does not check out, you made an arithmetic error.