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Bearing Capacity of Soil — Terzaghi's Theory with 3 Worked Examples

⏱ 18 min read📅 June 2026🎓 GATE relevant
Bearing capacity is the single most important parameter in foundation engineering — it tells you how much load the soil can safely carry per unit area. Terzaghi (1943) gave the first rigorous theoretical formula for bearing capacity based on a failure mechanism involving three distinct zones beneath the footing. This guide derives the formula conceptually, explains the three bearing capacity factors (Nc, Nq, Nγ), covers general vs local shear failure, and includes three worked examples and 10 GATE MCQs.

📋 Table of Contents

  1. Introduction
  2. Concept and Theory
  3. IS Code Background
  4. Key Formulas
  5. Bearing Capacity Factor Tables
  6. Calculation Procedure
  7. Worked Examples (3)
  8. GATE MCQs (10)
  9. Common Mistakes
  10. Revision Summary
  11. Related Articles

1. Introduction

When you place a footing on soil and gradually increase the load, the soil eventually fails — it squeezes out sideways and the footing sinks. The load per unit area at which this happens is called the ultimate bearing capacity (qu). The safe bearing capacity (qsafe) is qu divided by a factor of safety (typically 2.5 to 3.0), giving the maximum pressure you can safely apply through the footing.

Terzaghi's equation is the foundation of all bearing capacity theory. Every modern bearing capacity formula (Meyerhof, Hansen, Vesic) is an extension of Terzaghi's original work with shape factors, depth factors, and inclination factors added on top.

2. Concept and Theory

The three failure zones

Terzaghi assumed that when a strip footing fails, the soil beneath it forms three distinct zones: (1) a rigid active zone (triangular wedge directly under the footing that moves down with the footing), (2) a radial shear zone (fan-shaped zone where soil is being pushed sideways, shear surfaces are log spirals), and (3) a passive zone (triangular wedge on each side that is pushed upward and outward). The bearing capacity is the load required to push the active wedge down against the resistance of these three zones.

Three components of bearing capacity

The formula has three additive terms, each representing a different source of soil resistance: cohesion (c) — the glue-like bond between soil particles (important in clay); surcharge (q) — the weight of soil above the footing level, which acts like a confining pressure; and self-weight (½γBNγ) — the weight of the soil in the failure zone itself, which resists being pushed up. Each term has its own bearing capacity factor (Nc, Nq, Nγ) that depends only on the soil's angle of internal friction φ.

General vs local shear failure

General shear failure occurs in dense/stiff soils — the failure surface extends all the way to the ground surface with clear heaving visible on both sides of the footing. Local shear failure occurs in loose/soft soils — the failure is internal, the soil compresses significantly, and no clear failure surface reaches the ground. For local shear, Terzaghi recommends using reduced values: c' = 2c/3 and tan φ' = 2 tan φ/3 (i.e., reduce both cohesion and friction angle).

3. IS Code Background

CodeSubjectPlain English
IS 6403Bearing capacity (shallow foundations)Gives Terzaghi's and Meyerhof's formulae with shape, depth, and inclination factors. Factor of safety = 2.5 to 3.0 for SBC.
IS 1904Foundation design — general requirementsSpecifies depth of foundation ≥ 0.5m for normal soils. Prescriptive SBC values for different soil types when detailed investigation is not available.
IS 2720Soil testsStandardises tests to determine c, φ, γ — the inputs to the bearing capacity formula.

4. Key Formulas

Terzaghi's Bearing Capacity — Strip Footing (General Shear)
qu = c × Nc + q × Nq + 0.5 × γ × B × Nγ

qu = ultimate bearing capacity (kN/m²)
c = cohesion of soil (kN/m²)
q = overburden pressure at footing level = γ × Df
γ = unit weight of soil (kN/m³)
B = width of footing (m)
Df = depth of footing below ground (m)
Nc, Nq, Nγ = bearing capacity factors (depend on φ only)
Shape Corrections (Terzaghi)
Square footing: qu = 1.3cNc + qNq + 0.4γBNγ
Circular footing: qu = 1.3cNc + qNq + 0.3γBNγ
(B = diameter for circular footing)
Local Shear Failure Modification
Replace c with c' = 2c/3
Replace φ with φ' where tan φ' = (2/3) tan φ
Use Nc', Nq', Nγ' from tables using φ'
Safe Bearing Capacity
qsafe = qu / FOS (gross)
qnet,safe = (qu − q) / FOS + q (net + overburden)
FOS = 2.5 to 3.0 typically

5. Bearing Capacity Factor Tables

φ (°)NcNqNγ
05.71.00.0
57.31.60.5
109.62.71.2
1512.94.42.5
2017.77.45.0
2525.112.79.7
3037.222.519.7
3557.841.442.4
4095.781.3100.4

How to use: From soil test, get c and φ. Look up Nc, Nq, Nγ for your φ. Substitute into the formula. The factors increase exponentially with φ — a change from 30° to 35° nearly doubles the capacity.

6. Calculation Procedure

  1. Determine soil parameters: c, φ, γ from lab tests (triaxial, direct shear, SPT correlations).
  2. Decide failure type: Dense/stiff → general shear (use c, φ directly). Loose/soft → local shear (use 2c/3, modified φ).
  3. Look up N factors from the table for your φ.
  4. Calculate overburden: q = γ × Df.
  5. Apply Terzaghi's formula with appropriate shape correction (strip, square, circular).
  6. Divide by FOS to get safe bearing capacity.

7. Worked Examples

Example 1 — Strip Footing on Cohesive Soil (Beginner)
A strip footing 1.5m wide at 1.0m depth. Soil: c = 25 kN/m², φ = 0° (saturated clay), γ = 18 kN/m³. FOS = 3.0.
Step 1 — Bearing Capacity Factors
φ = 0°: Nc = 5.7, Nq = 1.0, Nγ = 0
Step 2 — Overburden
q = 18 × 1.0 = 18 kN/m²
Step 3 — Ultimate Bearing Capacity
qu = 25 × 5.7 + 18 × 1.0 + 0.5 × 18 × 1.5 × 0 = 142.5 + 18 + 0 = 160.5 kN/m²
Step 4 — Safe Bearing Capacity
qsafe = 160.5 / 3.0 = 53.5 kN/m²
Example 2 — Square Footing on Sandy Soil (Intermediate)
Square footing 2m × 2m at 1.5m depth. Soil: c = 0, φ = 30°, γ = 19 kN/m³. FOS = 2.5.
Step 1
φ = 30°: Nc = 37.2, Nq = 22.5, Nγ = 19.7
Step 2
q = 19 × 1.5 = 28.5 kN/m²
Step 3 — Square Footing Formula
qu = 1.3 × 0 × 37.2 + 28.5 × 22.5 + 0.4 × 19 × 2 × 19.7
= 0 + 641.25 + 299.4 = 940.7 kN/m²
Step 4
qsafe = 940.7 / 2.5 = 376.3 kN/m²
Example 3 — Local Shear Failure in Loose Sand (Advanced)
Strip footing 1.2m wide at 0.8m depth. Loose sand: c = 5 kN/m², φ = 25°, γ = 16 kN/m³. Local shear failure expected. FOS = 3.0.
Step 1 — Modified Parameters
c' = 2 × 5 / 3 = 3.33 kN/m²
tan φ' = (2/3) tan 25° = (2/3)(0.4663) = 0.3109 → φ' = 17.3°
Step 2 — N Factors for φ' = 17.3° (interpolated)
Nc' ≈ 14.6, Nq' ≈ 5.3, Nγ' ≈ 3.4
Step 3
q = 16 × 0.8 = 12.8 kN/m²
qu = 3.33 × 14.6 + 12.8 × 5.3 + 0.5 × 16 × 1.2 × 3.4
= 48.6 + 67.8 + 32.6 = 149.0 kN/m²
Step 4
qsafe = 149.0 / 3.0 = 49.7 kN/m²

8. GATE MCQs

Q1. Terzaghi's bearing capacity equation for a strip footing has how many terms?
  1. (a) 2
  2. (b) 3
  3. (c) 4
  4. (d) 1
Answer: (b)
Three terms: cohesion (cNc), surcharge (qNq), and self-weight (0.5γBNγ).
Q2. For purely cohesive soil (φ = 0), the Terzaghi bearing capacity factor Nc is:
  1. (a) 5.14
  2. (b) 5.7
  3. (c) 9.0
  4. (d) 1.0
Answer: (b)
Terzaghi gives Nc = 5.7 for φ = 0°. (Meyerhof gives 5.14 — don't mix the two.)
Q3. For local shear failure, cohesion is reduced to:
  1. (a) c/2
  2. (b) 2c/3
  3. (c) c/3
  4. (d) 3c/4
Answer: (b)
c' = 2c/3 and tan φ' = (2/3) tan φ for local shear failure. Memorise the 2/3 factor.
Q4. Terzaghi's shape factor for a square footing modifies the cohesion term to:
  1. (a) 1.0 × cNc
  2. (b) 1.2 × cNc
  3. (c) 1.3 × cNc
  4. (d) 1.5 × cNc
Answer: (c)
Square footing: 1.3cNc (cohesion term), 0.4γBNγ (self-weight term).
Q5. If φ increases from 20° to 30°, the bearing capacity factor Nγ approximately:
  1. (a) Doubles
  2. (b) Triples
  3. (c) Quadruples
  4. (d) Remains same
Answer: (c)
Nγ at 20° = 5.0, at 30° = 19.7. Ratio = 19.7/5.0 ≈ 4. The factors increase exponentially with φ.
Q6. The overburden pressure q in Terzaghi's formula is:
  1. (a) γ × B
  2. (b) γ × Df
  3. (c) γ × (B + Df)
  4. (d) c × Df
Answer: (b)
q = γ × Df, where Df is the depth of the footing below ground level. This surcharge confines the failure zone and increases capacity.
Q7. General shear failure is expected in:
  1. (a) Loose sand and soft clay
  2. (b) Dense sand and stiff clay
  3. (c) All soils at all densities
  4. (d) Only saturated soils
Answer: (b)
General shear = dense/stiff soils with well-defined failure surface. Local shear = loose/soft soils with gradual failure.
Q8. For φ = 0 (pure clay), the Nγ factor is:
  1. (a) 5.7
  2. (b) 1.0
  3. (c) 0
  4. (d) 0.5
Answer: (c)
Nγ = 0 when φ = 0. The self-weight term vanishes for purely cohesive soil — only cohesion and surcharge contribute.
Q9. A typical factor of safety for bearing capacity is:
  1. (a) 1.5
  2. (b) 2.0
  3. (c) 2.5 to 3.0
  4. (d) 5.0
Answer: (c)
IS 6403 recommends FOS of 2.5 for normal cases and 3.0 for important structures.
Q10. Increasing the footing depth Df increases bearing capacity primarily because:
  1. (a) The footing gets stronger
  2. (b) The surcharge q = γDf increases, adding to the qNq term
  3. (c) The soil gets stronger with depth
  4. (d) Water table goes deeper
Answer: (b)
Greater depth means more overburden pressure q, which multiplied by Nq directly adds to bearing capacity. The surcharge confines the failure zone.

9. Common Mistakes

Mistake 1: Using Meyerhof's Nc = 5.14 in Terzaghi's formula. Terzaghi gives Nc = 5.7 for φ = 0. Meyerhof gives 5.14. Don't mix formulas and factors from different authors.
Mistake 2: Forgetting shape factors for square/circular footings. The standard formula is for strip footings. Square and circular need the 1.3/0.4 and 1.3/0.3 corrections respectively.
Mistake 3: Not reducing c and φ for local shear. If the soil is loose or soft, you must use c' and φ', not the original values.
Mistake 4: Confusing gross and net bearing capacity. qnet = qu − q (subtract overburden). For net SBC: qnet,safe = (qu − q)/FOS. The difference matters for design.

10. Quick Revision Summary

Memorise:

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