🎓 Lesson 11
D5
Stress Attenuation Laws: Boussinesq, Westergaard, and Numerical Calibration
Stress attenuation laws tell us how much pressure from a surface load—like a tire or blast—weakens as it spreads downward and outward into the ground.
🎯 Learning Objectives
- ✓ Calculate vertical stress at any depth and radial distance using Boussinesq and Westergaard equations
- ✓ Analyze the influence of Poisson’s ratio and layering on stress distribution by comparing Boussinesq vs. Westergaard predictions
- ✓ Apply numerical calibration techniques (e.g., back-analysis of pressure sensor data) to adjust analytical models for field-observed compaction depth
- ✓ Explain why Boussinesq overestimates near-surface stress but underestimates depth of influence in stiff, layered soils
- ✓ Design tire–soil contact pressure models that integrate analytical attenuation laws with measured soil modulus profiles
📖 Why This Matters
In mining haulage, tire–soil interaction governs rutting, subgrade failure, and fuel efficiency—especially on unsealed haul roads and stockpiles. Overestimating stress attenuation leads to undersized road sections and premature compaction; underestimating it causes unnecessary overdesign and cost. Understanding how pressure fades with depth isn’t academic—it directly determines safe axle loads, optimal tire inflation, and sustainable compaction depth limits for long-term pit infrastructure.
📘 Core Principles
Boussinesq’s solution assumes a point load on an infinite, homogeneous, isotropic, linearly elastic half-space with Poisson’s ratio ν = 0.5—ideal for soft, uniform soils. Westergaard modifies this for thin, stiff surface layers over softer substrates (e.g., crusty mine haul roads), assuming ν ≈ 0 and vertical restraint—yielding sharper stress concentration near the surface but faster lateral decay. Real soils violate both assumptions: they’re nonlinear, anisotropic, and stratified. Hence, numerical calibration—using field-measured vertical strain or pressure cells at multiple depths—is essential to scale analytical stress predictions and define the effective compaction depth (typically where σ_z < 10% of surface contact pressure).
📐 Key Calculation
The Boussinesq equation computes vertical stress σ_z beneath a point load; for distributed contact (e.g., tire footprint), it’s integrated over area. The Westergaard solution replaces the denominator’s geometric term to reflect constrained lateral strain. Both are foundational for vertical pressure gradient modeling in Module 6.
Boussinesq Vertical Stress (Point Load)
σ_z = (3P / 2π) × (z³ / (r² + z²)^(5/2))Vertical normal stress at depth z and radial distance r from a concentrated vertical load P on an elastic half-space.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| σ_z | Vertical stress | Pa | Compressive stress acting perpendicular to horizontal plane at (r,z) |
| P | Applied point load | N | Concentrated vertical force (e.g., equivalent single axle load) |
| r | Radial distance from load axis | m | Horizontal offset from centerline of applied load |
| z | Depth below surface | m | Vertical distance from surface to calculation point |
Typical Ranges:
Mining haul tire (120 kN, r=0): 80–140 kPa at z=0.2 m
Same load, z=1.0 m: 5–12 kPa
💡 Worked Example
Problem: A rigid wheel applies a 120 kN point load on a crushed rock haul road (assumed homogeneous elastic medium). Calculate vertical stress at r = 0.4 m, z = 0.6 m.
1.
Step 1: Identify knowns — P = 120,000 N, r = 0.4 m, z = 0.6 m
2.
Step 2: Compute R = √(r² + z²) = √(0.16 + 0.36) = √0.52 ≈ 0.721 m
3.
Step 3: Apply Boussinesq: σ_z = (3P / 2π) × (z³ / R⁵) = (3×120000 / 2π) × (0.6³ / 0.721⁵) ≈ (57295.8) × (0.216 / 0.198) ≈ 57295.8 × 1.091 ≈ 62,500 Pa
4.
Step 4: Verify against typical range — For haul road subgrades, σ_z at z = 0.6 m under 120 kN is commonly 45–75 kPa; 62.5 kPa falls within expected bounds.
Answer:
The vertical stress is 62.5 kPa, which falls within the typical range of 45–75 kPa for this loading and depth.
🏗️ Real-World Application
At BHP’s South Flank iron ore operation (Pilbara, WA), pressure transducers embedded at 0.2, 0.4, 0.8, and 1.2 m depths beneath a 63/80R63 ultra-large mining tire revealed peak σ_z = 115 kPa at surface, decaying to 10.8 kPa at 1.2 m. Boussinesq predicted 13.2 kPa at 1.2 m (overestimating by 22%), while Westergaard predicted 9.4 kPa (underestimating by 13%). A calibrated FEM model—using in-situ CPT-derived E_s(z) and a ν(z) profile—reproduced measurements within ±4%, establishing an effective compaction depth of 1.15 m (where σ_z < 10% of surface pressure). This calibrated depth now informs haul road recompaction cycles and moisture control protocols.
🔧 Interactive Calculator
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