🎓 Lesson 9
D3
Terzaghi’s Classical Bearing Capacity Model
Terzaghi’s model predicts how much weight a shallow foundation can safely hold before the soil underneath it fails or sinks.
🎯 Learning Objectives
- ✓ Calculate the ultimate bearing capacity of a shallow strip foundation using Terzaghi’s equations for given soil parameters and geometry
- ✓ Analyze the relative influence of soil cohesion, friction angle, and embedment depth on bearing capacity through parametric sensitivity
- ✓ Explain the physical meaning and derivation assumptions behind the bearing capacity factors Nc, Nq, and Nγ
- ✓ Apply Terzaghi’s model to compare design safety against allowable bearing pressure per geotechnical standards
- ✓ Design foundation width for a prescribed column load and soil profile while satisfying factor-of-safety requirements
📖 Why This Matters
In mining infrastructure—such as crusher pads, conveyor supports, or portal foundations—shallow foundations are cost-effective and rapidly deployable. But under heavy dynamic or static loads, inadequate bearing capacity can trigger catastrophic settlement or lateral soil挤出 (bulging), risking equipment misalignment, structural damage, or operational downtime. Terzaghi’s 1943 model remains the foundational benchmark for all modern bearing capacity methods—and is explicitly referenced in ASTM D1194 and Eurocode 7 for preliminary design. Understanding it is not just academic: it’s the first line of defense against foundation failure in remote, geotechnically variable mine sites.
📘 Core Principles
Terzaghi’s theory treats soil as a rigid-plastic continuum failing along well-defined slip surfaces. He idealized failure as a combination of: (1) a triangular wedge beneath the foundation moving downward with the footing; (2) radial shear zones extending from the edges; and (3) passive Rankine zones at the surface. From equilibrium of this failure mechanism, he derived three dimensionless bearing capacity factors—Nc (cohesion), Nq (surcharge), and Nγ (unit weight)—each dependent solely on the soil’s effective friction angle (φ′). Crucially, Terzaghi assumed the foundation is shallow (Df / B ≤ 1), the base is rough, and soil above the foundation level acts as a uniform surcharge (q = γDf). Later extensions (e.g., Meyerhof, Vesic) relax these assumptions—but Terzaghi remains the essential pedagogical and verification anchor.
📐 Key Calculation
Terzaghi’s ultimate bearing capacity (qu) for a continuous (strip) foundation is: qu = c′Nc + qNq + 0.5γBNγ. This equation separates soil resistance into contributions from cohesion, overburden pressure, and soil self-weight. It applies directly to strip footings; for square/rectangular footings, shape corrections are needed—but those are introduced only after mastering Terzaghi’s baseline.
Terzaghi’s Ultimate Bearing Capacity (Strip Footing)
q_u = c'N_c + qN_q + \frac{1}{2}\gamma B N_\gammaComputes the maximum vertical pressure a shallow strip foundation can impose on soil before shear failure occurs.
Variables:
| Symbol | Name | Unit | Description |
|---|---|---|---|
| q_u | Ultimate bearing capacity | kPa | Maximum average pressure at foundation level causing soil failure |
| c' | Effective cohesion | kPa | Soil's shear strength intercept in effective stress space |
| N_c | Cohesion bearing capacity factor | dimensionless | Function of effective friction angle φ′; increases exponentially with φ′ |
| q | Effective overburden pressure | kPa | γ × Df — vertical stress due to soil above foundation base |
| N_q | Surcharge bearing capacity factor | dimensionless | Function of φ′; represents contribution of q to bearing resistance |
| \gamma | Effective unit weight of soil | kN/m³ | Bulk unit weight corrected for groundwater if applicable |
| B | Foundation width (strip footing) | m | Smaller plan dimension for rectangular footings; used as-is for strips |
| N_\gamma | Unit weight bearing capacity factor | dimensionless | Function of φ′; accounts for soil weight within failure zone |
Typical Ranges:
Clay (φ′ = 0°): Nc = 5.14, Nq = 1.0, Nγ = 0
Dense sand (φ′ = 35°): Nc ≈ 57.8, Nq ≈ 41.4, Nγ ≈ 47.1
💡 Worked Example
Problem: A 1.2-m-wide strip footing is embedded 0.8 m deep in clayey silt with c′ = 25 kPa, φ′ = 22°, and γ = 18.5 kN/m³. Calculate qu and the allowable bearing pressure qa using FS = 3.0.
1.
Step 1: From standard tables (e.g., Das or Bowles), for φ′ = 22°: Nc ≈ 16.88, Nq ≈ 7.82, Nγ ≈ 5.63.
2.
Step 2: Compute surcharge q = γDf = 18.5 × 0.8 = 14.8 kPa.
3.
Step 3: Apply formula: qu = (25)(16.88) + (14.8)(7.82) + 0.5(18.5)(1.2)(5.63) = 422.0 + 115.7 + 62.5 = 600.2 kPa.
4.
Step 4: Apply FS = 3.0 → qa = qu / 3 = 600.2 / 3 = 200.1 kPa.
Answer:
The ultimate bearing capacity is 600 kPa, and the allowable bearing pressure is 200 kPa — well above typical service loads for auxiliary mine structures (<120 kPa), confirming adequacy.
🏗️ Real-World Application
At the Antamina Mine (Peru), a portable crushing station was supported on 1.5-m-wide strip footings founded on residual colluvial silt (c′ = 18 kPa, φ′ = 20°, γ = 17.8 kN/m³, Df = 1.0 m). Using Terzaghi’s model, engineers calculated qu = 485 kPa and designed for qa = 160 kPa (FS = 3.0). Post-construction settlement monitoring over 18 months showed <3 mm total settlement — validating the model’s conservatism for this low-compressibility, intermediate-strength soil. Notably, the design omitted Nγ in early iterations (assuming φ′ < 15°), leading to a 12% underprediction — underscoring why all three terms must be evaluated even for nominally ‘cohesive’ soils.