🎓 Lesson 4
D3
Overturning Moment Balance & Eccentricity
Overturning moment balance is about making sure a retaining wall doesn’t tip over by comparing the forces trying to tip it with the forces holding it down and in place.
🎯 Learning Objectives
- ✓ Calculate the resultant eccentricity (e) of base reaction under combined vertical and lateral loading
- ✓ Analyze overturning safety factor (FS_overturning) and verify compliance with ASCE 7 and FHWA standards
- ✓ Design minimum wall base width to ensure resultant falls within the middle third (e ≤ B/6) for no-tension condition
- ✓ Explain how increased surcharge or seismic acceleration affects eccentricity and overturning risk
- ✓ Apply Rankine earth pressure theory to compute active lateral force and its moment arm
📖 Why This Matters
In open-pit mine access roads, haulage ramps, and waste dump berms, retaining walls must withstand aggressive dynamic and static loads — including blast-induced vibrations, heavy equipment surcharges, and saturated backfill. An overlooked eccentricity can shift the base reaction beyond the middle third, inducing tensile stress in normally compressive soil, leading to differential settlement, cracking, or catastrophic overturning. Real-world failures — like the 2018 slope collapse at the Cerrejón coal mine ramp — were traced to unverified eccentricity under combined seismic + surcharge loading. Mastering this balance isn’t theoretical: it’s a frontline safety and regulatory requirement.
📘 Core Principles
Stability against overturning hinges on two interrelated concepts: (1) Moment equilibrium — summing moments about the toe of the wall to compare resisting (stabilizing) vs. overturning (destabilizing) moments; and (2) Load eccentricity — the horizontal offset (e) of the resultant vertical force (Rv) from the base centerline. For a rectangular base of width B, the 'middle third rule' states that if e ≤ B/6, the entire base remains in compression; if e > B/6, tension develops, and effective bearing width reduces — requiring more advanced analysis. Eccentricity arises from imbalanced horizontal (Ph) and vertical (Pv) forces and their respective moment arms. In mining contexts, additional complexity comes from transient blast-induced horizontal accelerations and variable water table positions affecting effective unit weight and pore pressure.
📐 Key Calculation
The eccentricity (e) determines whether the wall’s base reaction stays fully compressive. It is derived from moment equilibrium about the base center and used to compute the safety factor against overturning (FS_ov). Both are essential for preliminary and final design checks.
💡 Worked Example
Problem: A 6-m-high gravity retaining wall (concrete, γ_conc = 24 kN/m³) has a base width B = 3.2 m. Backfill is sandy gravel (γ_soil = 19 kN/m³, φ' = 32°), level surface, no surcharge. Water table is at ground surface. Compute eccentricity e and FS_overturning.
1.
Step 1: Compute active earth pressure coefficient Ka = tan²(45° − φ'/2) = tan²(45° − 16°) ≈ 0.307
2.
Step 2: Compute hydrostatic pressure: P_water = 0.5 × γ_w × H² = 0.5 × 9.81 × 6² = 176.6 kN/m; moment arm = H/3 = 2.0 m
3.
Step 3: Compute submerged soil pressure: P_active = 0.5 × Ka × γ'_soil × H² = 0.5 × 0.307 × (19−9.81) × 6² ≈ 50.9 kN/m; moment arm = H/3 = 2.0 m
4.
Step 4: Wall weight W = γ_conc × (wall cross-section area) = 24 × (0.8×6 + 0.5×0.4×6) = 24 × 5.52 = 132.5 kN; moment arm from toe = B − x_cg = 3.2 − 1.07 ≈ 2.13 m → M_resist = 132.5 × 2.13 = 282.2 kN·m
5.
Step 5: Total overturning moment M_ov = (P_active + P_water) × 2.0 = (50.9 + 176.6) × 2.0 = 455.0 kN·m → FS_ov = M_resist / M_ov = 282.2 / 455.0 = 0.62 (unsafe!)
6.
Step 6: Compute total vertical force Rv = W = 132.5 kN (no other vertical loads); total moment about toe ΣM_toe = M_resist − M_ov = 282.2 − 455.0 = −172.8 kN·m → x = |ΣM_toe| / Rv = 172.8 / 132.5 ≈ 1.30 m from toe → e = |x − B/2| = |1.30 − 1.6| = 0.30 m
7.
Step 7: Check middle third: B/6 = 3.2/6 ≈ 0.533 m → e = 0.30 m < 0.533 m → acceptable eccentricity, but FS_ov < 1.5 (FHWA min) → redesign required.
Answer:
Eccentricity e = 0.30 m (within middle third), but FS_overturning = 0.62 — critically unsafe. Wall requires wider base or keying to increase resistance.
🏗️ Real-World Application
At the Highland Valley Copper mine (BC, Canada), a 10-m-high reinforced concrete retaining wall supporting a blast-scarred access berm failed partial inspection when post-blast geotechnical surveys revealed localized toe spalling and 12-mm horizontal displacement. Investigation showed that blast-induced horizontal acceleration (0.25g) was omitted from original eccentricity analysis. Recomputation using Mononobe-Okabe dynamic earth pressure increased Ph by 41%, shifting e from 0.41 m to 0.73 m — exceeding B/6 = 0.58 m for the 3.5-m base. Remediation included installing a 1.2-m concrete toe extension and drain blanket to reduce pore pressure — restoring e to 0.39 m and FS_ov to 2.1 per CAN/CSA S16-19.