Bearing Capacity Factors — Nc, Nq, Nγ
Ultimate bearing capacity of a shallow foundation resolves into three contributions — cohesion, surcharge, and the self-weight of the failure wedge — each scaled by a dimensionless factor that depends only on the friction angle φ. Two families are in common use and they are not interchangeable: Terzaghi's original 1943 factors, and the Meyerhof/Vesić "general" factors that most modern codes build on.
Terzaghi's Equations
where q = γ·Df is the effective surcharge at founding level, B is the footing width (or diameter), and c is cohesion. The 1.3 / 0.4 / 0.3 multipliers are Terzaghi's empirical shape allowances, already built into the equations above — do not apply separate shape factors on top of them.
Terzaghi Bearing Capacity Factors
| φ (deg) | Nc | Nq | Nγ | Typical soil |
|---|---|---|---|---|
| 0 | 5.70 | 1.00 | 0.00 | saturated clay, undrained |
| 5 | 7.34 | 1.64 | 0.14 | soft silty clay |
| 10 | 9.61 | 2.69 | 0.56 | silt, clayey silt |
| 15 | 12.86 | 4.45 | 1.52 | loose silty sand |
| 20 | 17.69 | 7.44 | 3.64 | loose sand |
| 25 | 25.13 | 12.72 | 8.34 | medium sand |
| 30 | 37.16 | 22.46 | 19.13 | medium–dense sand |
| 35 | 57.75 | 41.44 | 45.41 | dense sand, gravel |
| 40 | 95.66 | 81.27 | 115.31 | very dense sand / gravel |
| 45 | 172.28 | 173.28 | 325.34 | dense angular gravel |
| 50 | 347.50 | 415.14 | 1072.80 | rarely justified from field data |
Interpolate linearly between rows for intermediate φ only where the interval is small; Nγ grows faster than linearly above φ = 35°, so prefer the closed-form expressions or the calculator there.
Meyerhof / Vesić General Factors
| φ (deg) | Nc | Nq | Nγ (Vesić) |
|---|---|---|---|
| 0 | 5.14 | 1.00 | 0.00 |
| 5 | 6.49 | 1.57 | 0.45 |
| 10 | 8.35 | 2.47 | 1.22 |
| 15 | 10.98 | 3.94 | 2.65 |
| 20 | 14.83 | 6.40 | 5.39 |
| 25 | 20.72 | 10.66 | 10.88 |
| 30 | 30.14 | 18.40 | 22.40 |
| 35 | 46.12 | 33.30 | 48.03 |
| 40 | 75.31 | 64.20 | 109.41 |
| 45 | 133.88 | 134.88 | 271.76 |
| 50 | 266.89 | 319.07 | 762.89 |
Closed-Form Expressions
| Factor | Terzaghi | Meyerhof / Vesić |
|---|---|---|
| Nq | e2(3π/4 − φ/2)tanφ / (2cos²(45° + φ/2)) | eπtanφ · tan²(45° + φ/2) |
| Nc | (Nq − 1)·cotφ | (Nq − 1)·cotφ |
| Nγ | tabulated (wedge solution) | 2(Nq + 1)·tanφ (Vesić) |
Nc is indeterminate at φ = 0 by the cotφ form; the limiting values are 5.70 (Terzaghi) and 5.14 (general). Meyerhof's own Nγ = (Nq − 1)tan(1.4φ) differs from Vesić's and runs lower at high φ — state which one you used.
Allowable Capacity and Factor of Safety
| Quantity | Expression | Note |
|---|---|---|
| Gross allowable | qall = qu / FS | FS = 3 typical for shallow foundations |
| Net ultimate | qu,net = qu − q | subtract the surcharge already there |
| Net allowable | qall,net = (qu − q) / FS | the value to compare against net applied pressure |
Sources: Terzaghi, K. (1943). Theoretical Soil Mechanics. Meyerhof, G.G. (1963). "Some recent research on the bearing capacity of foundations," Canadian Geotechnical Journal. Vesić, A.S. (1973). "Analysis of ultimate loads of shallow foundations," JSMFD ASCE. Das, B.M., Principles of Foundation Engineering — tabulated factor values. Verify φ against actual site investigation data rather than the "typical soil" column, which is orientation only.
Related cheat sheets and tools
Getting φ and c right matters more than the factor table — start from USCS classification and unit weights, and see presumptive bearing values for the code-table shortcut where a full analysis isn't warranted. For retaining and embankment work see lateral earth pressure and slope stability. For dam and basin embankment design that ties the geotechnical section to the hydraulics, see HydroComplete, the SaaS sister product to PE-Calc.