Bearing capacity is the ability of soil to support loads from a foundation without shear failure or excessive settlement. It is one of the first numbers a structural engineer asks for, because it sets the size of footings and decides whether a shallow foundation is feasible. This article explains the key definitions, the classical calculation methods, practical field correlations and a worked example you can follow.
Key Definitions
Ultimate bearing capacity (qu): the maximum pressure the soil can carry before shear failure occurs.
Net ultimate bearing capacity (qnu): qu minus the overburden pressure at foundation level (qnu = qu − γDf).
Safe bearing capacity (qs): net ultimate capacity divided by a factor of safety, plus overburden.
Allowable bearing pressure (qa): the pressure that satisfies both shear failure and settlement limits. In practice, settlement often governs, especially on sands and soft clays.
Modes of Bearing Failure
Terzaghi described three failure modes. General shear failure occurs in dense sands and stiff clays: a clear failure surface develops, the ground bulges at the sides and a peak load is reached. Local shear failure happens in medium-dense soils, with partial failure surfaces and no distinct peak. Punching shear failure occurs in very loose or soft soils, where the foundation sinks with little surface bulging. The classical equations are derived for general shear, and adjustments are used for the other cases.
Terzaghi’s Bearing Capacity Equation
Karl Terzaghi’s 1943 equation for a continuous strip footing under general shear is:
qu = c Nc + q Nq + 0.5 γ B Nγ
Where:
- c = cohesion of the soil (kPa)
- q = γ Df, the overburden pressure at foundation level (kPa)
- γ = unit weight of soil (kN/m³)
- B = width of footing (m)
- Nc, Nq, Nγ = dimensionless bearing capacity factors that depend only on friction angle φ
The three terms represent the contributions of cohesion, surcharge (the soil above the base) and the weight of soil below the base. For square footings, Terzaghi proposed qu = 1.3 c Nc + q Nq + 0.4 γ B Nγ, and for circular footings qu = 1.3 c Nc + q Nq + 0.3 γ B Nγ.
Typical Terzaghi factors (general shear): for φ = 0°, Nc = 5.7, Nq = 1.0, Nγ = 0; for φ = 20°, Nc = 17.7, Nq = 7.4, Nγ = 5.0; for φ = 30°, Nc = 37.2, Nq = 22.5, Nγ = 19.7; for φ = 40°, Nc = 95.7, Nq = 81.3, Nγ = 100.4. Note how quickly capacity rises with friction angle, which is why small errors in φ can cause large errors in capacity.
For local shear failure, the cohesion and friction angle are reduced: c′ = 2c/3 and tan φ′ = (2/3) tan φ.
Other Methods: Meyerhof, Hansen and Vesic
Later researchers improved on Terzaghi’s work. Meyerhof (1963) introduced shape, depth and inclination factors and a more realistic failure surface that extends into the soil above the footing base. Hansen (1970) and Vesic (1973) refined these factors further. The general form is:
qu = c Nc sc dc ic + q Nq sq dq iq + 0.5 γ B Nγ sγ dγ iγ
where s, d and i are shape, depth and load inclination factors. Eurocode 7 and many national codes follow this general approach and apply partial factors rather than a single global factor of safety.
Effect of Groundwater
Water in the soil reduces effective stress and therefore bearing capacity. If the water table is at or above foundation level, use the submerged unit weight γ′ = γsat − γw in the third term and for the overburden term as appropriate. If the water table is at depth B below the base or deeper, its effect on capacity is usually small. Capillary rise and seasonal fluctuations should be considered conservatively.
Factor of Safety
A factor of safety (FS) covers uncertainty in soil properties, theory, loads and construction. A value of 3.0 is commonly applied to ultimate bearing capacity for permanent structures under normal loading, and 2.0 or lower is sometimes accepted for temporary conditions or combined loading. Allowable pressure is then:
qa = qnu / FS
Under limit state design (Eurocode 7), the design resistance is the characteristic resistance divided by partial factors, and the design must satisfy both the ultimate limit state (failure) and serviceability limit state (settlement).
Worked Example: Square Footing on Sand
A 1.5 m by 1.5 m square footing is placed 1.0 m below ground level in dry sand. Soil properties: c = 0, φ = 30°, γ = 18 kN/m³. Find the gross ultimate and allowable bearing capacity using Terzaghi’s equation with FS = 3.
- Overburden q = γ Df = 18 × 1.0 = 18 kPa.
- For φ = 30°: Nq = 22.5, Nγ = 19.7.
- Square footing: qu = q Nq + 0.4 γ B Nγ (c = 0).
- qu = (18 × 22.5) + (0.4 × 18 × 1.5 × 19.7) = 405 + 212.8 ≈ 618 kPa.
- Gross allowable pressure qa = 618 / 3 ≈ 206 kPa.
The engineer must then check settlement. On sand, settlement under 206 kPa for a 1.5 m footing may be about 20 to 30 mm depending on density, and a settlement limit of 25 mm might reduce the allowable pressure further.
Using Field Tests
Standard Penetration Test (SPT). Meyerhof and others proposed charts and equations linking corrected SPT N-values to allowable bearing pressure for a settlement of 25 mm. For example, a simplified Meyerhof form for footings with B ≤ 1.2 m is qa (kPa) ≈ 12 N Kd, with Kd a depth factor, though modern practice uses refined, corrected correlations. SPT results must be corrected for energy, rod length, borehole diameter and overburden pressure.
Cone Penetration Test (CPT). Cone tip resistance qc correlates closely with bearing capacity and settlement in sands. A frequently used approximation for footings on sand relates allowable pressure to a fraction of qc, such as qa ≈ qc / 12 to qc / 15 for 25 mm settlement, depending on footing width.
Plate load test. A rigid steel plate, typically 300 to 750 mm square or circular, is loaded in stages in a test pit at foundation level (ASTM D1194, IS 1888). The load-settlement curve gives bearing pressure for a chosen settlement. Because the plate is small, scale effects are important: in sands, settlement increases with size, and in clays, the influence zone is shallow, so results may not represent deeper weak layers. Plate tests are best used as a check alongside borehole data.
Settlement Often Governs
In many cases, a footing that is safe against shear failure still settles too much. Sands usually settle quickly under load, while clays settle slowly through consolidation over months or years. Typical allowable settlements for isolated footings are 25 mm in sand and 50 mm in clay, with differential settlement limits of around 1/500 to 1/300 of span. Therefore, designers compare bearing pressure from the shear failure analysis with the pressure that causes limiting settlement and adopt the lower value.
Common Sources of Error
- Overestimating friction angle from poor correlations or untypical test samples.
- Ignoring weak layers below the footing. The influence zone extends roughly 1.5 to 2 B beneath the base, and a soft clay layer there can control behaviour (punching or layered failure).
- Neglecting groundwater rise after construction.
- Using undrained parameters for long-term conditions or the reverse.
- Applying equations outside their assumptions, for instance on slopes, near excavations or on eccentrically loaded footings (use effective width B′ = B − 2e).
Conclusion
Soil bearing capacity is calculated by combining soil strength parameters with foundation geometry, using Terzaghi’s equation or its modern refinements by Meyerhof, Hansen and Vesic. Field tests such as SPT, CPT and plate loading offer practical checks, while factors of safety and settlement limits convert theoretical capacity into allowable bearing pressure. Reliable values depend on good site investigation, careful interpretation and respect for groundwater and layered ground. When in doubt, a conservative approach and professional geotechnical advice protect both the structure and its users.
References and further reading
- Terzaghi, K. (1943). Theoretical Soil Mechanics
- Meyerhof, G.G. (1963). Some recent research on the bearing capacity of foundations. Canadian Geotechnical Journal
- Vesic, A.S. (1973). Analysis of ultimate loads of shallow foundations. Journal of the Soil Mechanics and Foundations Division, ASCE
- Das, B.M. Principles of Foundation Engineering
- EN 1997-1, Eurocode 7: Geotechnical design
- Bowles, J.E. Foundation Analysis and Design
