Table of Contents
1. Introduction to Bearing Capacity
Bearing capacity is the ability of soil to support the loads applied by a foundation without experiencing shear failure or excessive settlement. The ultimate bearing capacity (qult) is the maximum pressure the soil can sustain before failure. The allowable (safe) bearing capacity (qall) is qult divided by a factor of safety, typically 2.5-3.0, accounting for uncertainties in soil properties and loading.
Three modes of bearing capacity failure are recognized: general shear failure (sudden, catastrophic failure in dense sand or stiff clay), local shear failure (gradual failure with noticeable settlement), and punching shear failure (compression of soil under the footing with little lateral movement). The failure mode depends on soil density, relative compressibility, and foundation depth.
Foundation design requires that the applied bearing pressure does not exceed the allowable bearing capacity and that total and differential settlements remain within tolerable limits for the structure. The Soil Bearing Capacity Calculator automates these calculations for multiple methods.
2. Terzaghi's Bearing Capacity Theory
Karl Terzaghi developed the first comprehensive bearing capacity theory in 1943. His equation for a strip footing under vertical centric loading remains the foundation of modern bearing capacity analysis. Terzaghi considered a soil wedge beneath the footing that moves downward, forcing soil outward along curved slip surfaces. The contribution of cohesion, surcharge, and soil unit weight are represented by three bearing capacity factors Nc, Nq, and Nγ.
Terzaghi's bearing capacity factors are functions of the soil friction angle φ. For general shear failure: Nq = a²/(2cos²(45+φ/2)) where a = e^(0.75π - φ/2)tanφ, Nc = (Nq - 1)cotφ, and Nγ = (tanφ/2)(Kpγ/cos²φ - 1), where Kpγ is the passive earth pressure coefficient. These factors are tabulated in most geotechnical textbooks for φ values from 0 to 50 degrees.
For local shear failure conditions (loose sands or soft clays), Terzaghi recommended reducing c and φ to c' = ⅔c and φ' = arctan(⅔tanφ). This accounts for the progressive nature of failure in compressible soils.
3. Meyerhof and Hansen Methods
Meyerhof (1963) extended Terzaghi's work by considering the shear resistance along the failure surface above the footing base, incorporating depth factors and allowing for eccentric and inclined loads. His bearing capacity factors are similar but use a different Nγ expression: Nγ = (Nq - 1)tan(1.4φ). Meyerhof's method is widely used for shallow foundation design due to its simplicity and reasonable accuracy.
Hansen (1970) further refined the analysis by adding separate shape, depth, inclination, base tilt, and ground slope factors. His equation is: qult = cNc·sc·dc·ic·gc·bc + γDNq·sq·dq·iq·gq·bq + 0.5γBNγ·sγ·dγ·iγ·gγ·bγ. This comprehensive formulation is recommended for critical structures or complex loading conditions. Hansen's Nγ = 1.5(Nq - 1)tanφ for φ > 0.
Both methods are implemented in the Soil Bearing Capacity Calculator, which computes qult and qall for each method and highlights the governing value.
4. Shape, Depth, and Inclination Factors
Shape factors account for the geometry of the footing. For rectangular footings (B/L < 1), common shape factors per Hansen are: sc = 1 + (Nq/Nc)(B/L), sq = 1 + (B/L)sinφ, and sγ = 1 - 0.4(B/L). These factors transition between strip footing behavior (B/L = 0) and square footing behavior (B/L = 1).
Depth factors reflect the increased bearing capacity due to the shearing resistance of soil above the footing base. For Df/B ≤ 1: dc = 1 + 0.4(Df/B), dq = 1 + 2tanφ(1 - sinφ)²(Df/B), and dγ = 1.0. For Df/B > 1, the arctan(Df/B) in radians replaces the (Df/B) term. Inclination factors reduce capacity when the load is inclined from vertical, using the horizontal-to-vertical load ratio.
Ground slope factors (gc, gq, gγ) account for footing on or near slopes, and base tilt factors (bc, bq, bγ) account for inclined footing bases. These are essential for hillside construction or inclined foundation bearing surfaces.
5. Safe Bearing Capacity and Factor of Safety
The allowable bearing capacity is qall = qult / FS. The appropriate factor of safety depends on the reliability of soil parameters, the importance of the structure, and the consequence of failure. Typical minimum FS values: 3.0 for general shear failure (most structures), 2.5 for temporary structures, and 1.5-2.0 when using peak strength in undrained analysis for clays.
The net allowable bearing capacity (qnet) is often used for design: qnet = (qult - γDf) / FS, where γDf represents the overburden pressure at footing base level. This net value correctly accounts for the fact that the soil at foundation level is already supporting the weight of the overlying soil.
For preliminary design, presumptive bearing capacities from building codes can be used. Typical values: hard rock (10 MPa), stiff clay (150-300 kPa), dense sand (300-500 kPa), loose sand (100-200 kPa), and soft clay (50-100 kPa). These values must be verified by site-specific investigation for final design. The Footing Size Calculator uses bearing capacity to determine required footing dimensions.
6. Worked Example
Calculate Bearing Capacity for a Strip Footing
Given: Strip footing B = 1.5 m, depth Df = 1.0 m. Soil: c = 15 kPa, φ = 30°, γ = 18 kN/m³. Water table at great depth. Calculate qult using Terzaghi and the allowable bearing capacity with FS = 3.
Step 1: Bearing capacity factors for φ = 30° (Terzaghi, general shear). Nc = 37.2, Nq = 22.5, Nγ = 19.7.
Step 2: Apply Terzaghi's strip footing equation. qult = cNc + γDfNq + 0.5γBNγ = 15(37.2) + 18(1.0)(22.5) + 0.5(18)(1.5)(19.7) = 558 + 405 + 266 = 1229 kPa.
Step 3: Net ultimate bearing capacity. qnet(ult) = qult - γDf = 1229 - 18 = 1211 kPa.
Step 4: Allowable bearing capacity. qall = qnet(ult)/FS = 1211/3 = 404 kPa.
Check with Meyerhof: For φ = 30°, Nc = 30.1, Nq = 18.4, Nγ = 15.1 (Meyerhof). qult = 15(30.1) + 18(1)(18.4) + 0.5(18)(1.5)(15.1) = 452 + 331 + 204 = 987 kPa. qall = (987 - 18)/3 = 323 kPa. Meyerhof gives a more conservative result.
Use the Soil Bearing Capacity Calculator to compare all three methods.
Bearing Capacity Failure Modes
[SVG Diagram: Three bearing capacity failure modes illustrated side by side. General shear failure shows a continuous slip surface extending to the ground surface. Local shear shows a partial slip surface. Punching shear shows vertical compression directly under the footing with minimal lateral displacement.]
7. Frequently Asked Questions
What is the difference between ultimate and safe bearing capacity?
Ultimate bearing capacity (qult) is the maximum pressure the soil can support before shear failure. Safe (allowable) bearing capacity (qall) is qult divided by a factor of safety (typically 2.5-3.0) to provide a margin against failure and limit settlement to acceptable levels.
Which bearing capacity method should I use?
Terzaghi's method is suitable for most routine shallow foundation designs. Meyerhof's method is preferred when considering depth and inclination effects. Hansen's method is recommended for complex loading, inclined bases, or slopes. Comparing multiple methods provides confidence in the design value.
How does the water table affect bearing capacity?
A high water table reduces bearing capacity because the submerged unit weight (γsub = γsat - γw) replaces the total unit weight in the Nq and Nγ terms. If the water table is within B of the footing base, the unit weight must be reduced, sometimes by 50% or more.
What are presumptive bearing capacities?
Presumptive (allowable) bearing capacities are tabulated values published in building codes based on soil classification and visual description. They are used for preliminary design or minor structures but must be verified by geotechnical investigation for permanent or critical structures.
References & Standards
- Terzaghi, K. Theoretical Soil Mechanics. Wiley, 1943.
- Meyerhof, G.G. "Some Recent Research on the Bearing Capacity of Foundations." Canadian Geotechnical Journal, 1(1), 1963.
- Hansen, J.B. "A Revised and Extended Formula for Bearing Capacity." Danish Geotechnical Institute, Bulletin 28, 1970.
- Das, B.M. Principles of Foundation Engineering. 9th ed., Cengage Learning, 2020.
- IS 6403:1981. Code of Practice for Determination of Bearing Capacity of Shallow Foundations.
- Civil Engineering Handbook — Geotechnical Engineering chapter.
- Engineering Formula Library — Bearing capacity formulas.
- Engineering Glossary — Geotechnical definitions.