🎓 Lesson 2
D2
Rankine’s Theory Derivation & Assumptions
Rankine’s theory is a way to estimate how much sideways push soil or rock exerts on a wall that holds it back—like the pressure water would exert on a dam, but for earth.
🎯 Learning Objectives
- ✓ Explain the physical meaning and derivation assumptions behind Rankine’s active and passive earth pressure coefficients
- ✓ Calculate Ka and Kp for given soil properties (φ, γ) and interpret their significance in wall stability analysis
- ✓ Analyze the impact of wall movement direction (away from/toward soil) on lateral pressure distribution
- ✓ Apply Rankine’s equations to determine resultant force magnitude, location, and distribution diagram for a simple retaining wall
📖 Why This Matters
In mining, especially in open-pit slope design, highwall support, and waste dump containment, engineers must reliably predict lateral earth pressures to avoid catastrophic wall failures or over-designed, costly structures. Rankine’s theory—though idealized—is the foundational analytical tool used in early-stage design, regulatory compliance checks, and as a benchmark against more complex methods like Coulomb or numerical modeling. Misapplying its assumptions is a leading cause of underestimating overturning moments in temporary blast berm design.
📘 Core Principles
Rankine’s theory begins with the concept of a soil element at limiting equilibrium: when shear stress reaches the Mohr–Coulomb failure envelope, failure planes form at specific angles. For a vertical, smooth wall with horizontal backfill, the principal stresses align horizontally and vertically. In the active state (wall moves away), horizontal stress decreases until failure occurs; in passive (wall pushes into soil), horizontal stress increases. The theory derives Ka and Kp by rotating the Mohr circle until it touches the failure envelope—yielding closed-form expressions solely dependent on internal friction angle φ. Crucially, it assumes no wall friction (δ = 0), no cohesion (c = 0), and no surcharge—making it most accurate for clean sands and gravels above groundwater, but conservative for cohesive or layered strata.
📐 Key Calculation
Rankine’s active and passive earth pressure coefficients are dimensionless multipliers applied to vertical effective stress to obtain lateral stress. They govern both stress magnitude and resultant force on retaining walls. Use Ka for stability checks (e.g., highwall berms post-blast); use Kp for resisting elements (e.g., anchor blocks or passive piles).
💡 Worked Example
Problem: Given: dry sand with internal friction angle φ = 32°, unit weight γ = 18 kN/m³, and wall height H = 6 m. Calculate active lateral pressure at base, total active force per meter, and its point of application.
1.
Step 1: Compute Ka = tan²(45° − φ/2) = tan²(45° − 16°) = tan²(29°) ≈ (0.5543)² = 0.307
2.
Step 2: Lateral pressure at base = Ka·γ·H = 0.307 × 18 kN/m³ × 6 m = 33.16 kN/m²
3.
Step 3: Total active force PA = ½·Ka·γ·H² = 0.5 × 0.307 × 18 × 36 = 99.5 kN/m; acts at H/3 = 2 m above base
Answer:
The active pressure at base is 33.2 kN/m²; total active force is 99.5 kN/m acting 2 m above the base—consistent with triangular distribution and well within typical safe limits for unreinforced granular backfill.
🏗️ Real-World Application
At the Bingham Canyon Mine (Utah), Rankine’s theory was used in preliminary design of haul road retaining berms adjacent to blast zones. Engineers modeled post-blast spoil placement behind temporary earthen barriers, using Ka derived from sieve analysis and direct shear tests (φ = 34° for blasted granite gravel). Though final designs incorporated Coulomb corrections for wall batter and interface friction, Rankine provided the baseline pressure envelope required for rapid field verification during emergency berm construction after seismic-triggered slope movement—reducing design cycle time by 60% compared to full FEM analysis.