🎓 Lesson 2
D2
Soil Stress–Strain Behavior Under Dynamic Wheel Loads
How soil squishes and bounces back when heavy vehicle wheels roll over it, especially during rapid loading like mining haul truck maneuvers.
🎯 Learning Objectives
- ✓ Calculate vertical and shear stress distributions beneath a rolling tire using Boussinesq and elastic–viscoelastic models
- ✓ Analyze strain accumulation and recovery cycles to predict rut depth evolution under repeated dynamic loading
- ✓ Apply empirical tire–soil pressure distribution coefficients (e.g., k₁, k₂) to design haul road subgrade compaction specifications
- ✓ Explain the influence of soil moisture content, density, and frequency of loading on dynamic modulus reduction
📖 Why This Matters
In open-pit mines, haul trucks weighing up to 400 tonnes exert dynamic wheel loads that repeatedly deform haul road surfaces—causing rutting, aggregate degradation, and increased fuel consumption. Understanding how soil responds *during* wheel passage—not just under static load—is critical to designing durable, low-maintenance haul roads and avoiding unplanned downtime. Ignoring dynamic effects leads to over-designed (costly) or under-designed (unsafe) subgrades.
📘 Core Principles
Soil under dynamic wheel loads exhibits three key behaviors: (1) Time-dependent response—stress transmission lags strain due to pore fluid movement and particle rearrangement; (2) Nonlinearity—modulus decreases with increasing stress amplitude and moisture; (3) Hysteresis—energy loss per cycle manifests as heat and permanent deformation. The contact pressure is not uniform: it peaks near the leading edge and decays toward the trailing edge, influenced by tire inflation pressure, tread geometry, and soil stiffness. Dynamic loading amplifies peak stresses by 1.3–2.5× compared to static equivalents, depending on speed and damping ratio.
📐 Dynamic Vertical Stress at Depth (Boussinesq–Viscoelastic Approximation)
This modified Boussinesq solution accounts for dynamic amplification and soil damping. It estimates peak vertical stress σ_z at depth z beneath the centerline of a circular loaded area (tire contact patch), incorporating a dynamic amplification factor (DAF) derived from field-calibrated vibration data.
Dynamic Vertical Stress (σ_z,dynamic)
σ_z,dynamic = DAF × p₀ × [1 / (1 + (z/a)²)^(3/2)]Estimates peak vertical stress at depth z beneath center of circular tire contact patch under dynamic loading.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| DAF | Dynamic Amplification Factor | dimensionless | Empirically calibrated multiplier reflecting speed, damping, and frequency effects |
| p₀ | Effective Contact Pressure | MPa | Average pressure over contact area, adjusted for tire–soil compliance (typically 0.7–0.9 × inflation pressure) |
| z | Depth Below Surface | m | Vertical distance from surface to point of interest |
| a | Contact Patch Radius | m | Radius of equivalent circular contact area (derived from measured footprint or manufacturer data) |
Typical Ranges:
Dry dense sand, 20 km/h: 1.3 – 1.6
Saturated silty clay, 40 km/h: 1.9 – 2.5
💡 Worked Example
Problem: A 63/80R63 mining tire operates at 7.5 bar inflation pressure, generating a circular contact patch of radius a = 0.32 m. Soil has a dynamic amplification factor DAF = 1.8 (measured at 30 km/h). Calculate σ_z at z = 0.6 m depth directly below the patch center.
1.
Step 1: Compute static contact pressure p₀ = inflation pressure × 0.85 (empirical effective pressure ratio) = 7.5 × 0.85 = 6.375 MPa.
2.
Step 2: Apply Boussinesq static solution: σ_z,static = p₀ × [1 / (1 + (z/a)²)^(3/2)] = 6.375 × [1 / (1 + (0.6/0.32)²)^(3/2)] = 6.375 × [1 / (1 + 3.516)^(1.5)] ≈ 6.375 × 0.105 = 0.670 MPa.
3.
Step 3: Multiply by DAF: σ_z,dynamic = 0.670 × 1.8 = 1.206 MPa.
Answer:
The dynamic vertical stress at 0.6 m depth is 1.21 MPa, which exceeds typical allowable subgrade stress (0.3–0.5 MPa for granular base), indicating need for improved compaction or thicker pavement section.
🏗️ Real-World Application
At Rio Tinto’s Pilbara iron ore operations, haul road rutting exceeded 120 mm/year on sections with clay-rich subgrade (LL = 42%, PI = 24%). Geotechnical monitoring revealed dynamic stress amplification >2.0× at 0.5 m depth during 35 km/h truck transit. By recalibrating compaction targets to achieve dynamic modulus (E_d) ≥ 180 MPa (measured via Falling Weight Deflectometer at 25 Hz), and adding a 150-mm stabilized capping layer, rutting was reduced by 68% over two wet seasons—demonstrating direct linkage between dynamic stress–strain modeling and operational cost savings.