Shallow Foundation Bearing Capacity Calculator

Introduction to shallow foundation bearing capacity analysis

A shallow foundation transfers structural load to the ground through a footing whose founding depth is comparable to, or smaller than, its own width. Before such a footing can be sized, the geotechnical engineer has to know the pressure at which the soil beneath it would develop a continuous shear failure surface and punch the footing into the ground. That pressure is the ultimate bearing capacity, q_ult. Dividing it by a factor of safety produces the allowable bearing pressure, q_all, which is the value a structural engineer compares against the contact stress delivered by the column or wall.

This calculator evaluates the general bearing capacity equation in the form published by the Federal Highway Administration in Geotechnical Engineering Circular No. 6 – Shallow Foundations (FHWA-SA-02-054), Equation 5-14, which is the same formulation adopted by the AASHTO LRFD Bridge Design Specifications. It handles strip, square, circular and rectangular footings, corrects for the position of the ground water table, and lets you switch between the four bearing capacity factor sets that dominate the literature: Terzaghi (1943), Meyerhof (1963), Brinch Hansen (1970) and Vesic (1973).

That last control matters more than it looks. A very large number of online bearing capacity tools print the Prandtl–Reissner factors under a heading that says “Terzaghi”. The two sets are genuinely different — at a friction angle of 30° they disagree by roughly 20 % on N_c and N_q and by 14 % in the opposite direction on N_γ — so this page always names the factor set it used and reproduces the published tabulated values exactly.

How to use this shallow foundation calculator step by step

1. Enter the drained or undrained strength parameters. Cohesion c is in kPa and the effective friction angle φ is in degrees. For an undrained (short term) check on saturated clay, enter c = s_u and φ = 0. For a drained (long term) check, enter the effective cohesion and effective friction angle.
2. Enter the soil unit weight γ in kN/m³. Use the total (bulk) unit weight; the calculator applies the ground water correction factors separately rather than asking you to pre-compute a buoyant weight.
3. Choose the footing shape and enter the width B in metres. B is always the shorter plan dimension, or the diameter of a circular footing. For a rectangular footing, also give the length L.
4. Enter the founding depth D_f, measured from the lowest adjacent finished grade to the underside of the footing, and the depth to the ground water table D_w measured from the same surface. Enter a water depth greater than 1.5B + D_f if there is no water table within the zone of influence.
5. Pick the factor set and the failure mode. Leave the factor set on the FHWA/AASHTO default unless you are deliberately reproducing a textbook Terzaghi calculation. Select local shear only for loose sands or soft, compressible clays.
6. Enter a factor of safety and press Compute bearing capacity. The result panel reports the factors, the correction factors, each of the three capacity terms, the gross and net ultimate capacity, and the gross and net allowable pressure. The chart underneath sweeps the footing width so you can size the footing directly, and the CSV button exports that sweep.

The bearing capacity formula behind this calculator

The starting point is the Prandtl–Reissner plasticity solution for a weightless soil carrying a uniform surcharge, extended by an empirical unit weight term. In the FHWA form (Equation 5-14 of Geotechnical Engineering Circular No. 6, with the base inclination and embedment depth factors taken as unity) the gross ultimate bearing pressure is:

The surcharge acting at founding level is simply the weight of the overburden that was removed:

Physically the three terms are the shear resistance mobilised by cohesion, the confinement provided by the overburden either side of the footing, and the resistance generated by the self-weight of the soil inside the failure wedge. The first two come from a rigorous limit-equilibrium solution; the third has no closed-form solution and is the reason different authors publish different N_γ values.

Bearing capacity factors N_c, N_q and N_γ

For the FHWA/AASHTO default, Meyerhof and Brinch Hansen options, the surcharge factor is the Reissner (1924) expression given as Equation 5-2 of Circular No. 6:

and the cohesion factor is the Prandtl (1920) expression of Equations 5-3 and 5-4, which is indeterminate at φ = 0 and must be replaced there by its limit:

The unit weight factor is where the published sets diverge. FHWA and AASHTO adopt the Caquot and Kerisel (1948) form, quoted as Equation 5-8 of Circular No. 6, which is numerically identical to the expression Vesic recommended in 1973:

Meyerhof and Brinch Hansen instead published, respectively:

Terzaghi’s 1943 solution is a different theory, not a different N_γ. It assumes a rough footing base, ignores the shear strength of the soil above founding level, and produces its own N_q from a log-spiral wedge whose half-angle equals φ:

Terzaghi’s N_γ has no accepted closed form — it was back-figured from a passive pressure coefficient read off a chart. This calculator therefore interpolates the tabulated Terzaghi values published in Table 4-1 of US Army Corps of Engineers manual EM 1110-1-1905, and labels them as tabulated in the output so you know they are not being generated from a formula.

Shape and ground water correction factors

For a footing whose length is less than five times its width, AASHTO requires shape corrections. With r = B/L (so r = 0 for a strip and r = 1 for a square or circle), Table 5-2 of Circular No. 6 gives, for φ > 0:

For φ = 0 the same table gives s_c = 1 + r/5 with s_q = s_γ = 1. When the Terzaghi factor set is selected the calculator switches to Terzaghi’s own shape corrections from Table 4-2 of EM 1110-1-1905 (strip 1.0/1.0, square 1.3/0.8, circular 1.3/0.6, with s_q = 1 throughout), because mixing one author’s factors with another author’s shape corrections is not defensible.

Water reduces the effective stress on the failure surface. Rather than asking you to compute a buoyant unit weight, Circular No. 6 applies two multipliers to the two unit weight terms, tabulated in Table 5-3:

Both are floored at 0.5. The published table anchors them at three points — water at the surface gives 0.5 and 0.5, water at founding level gives 1.0 and 0.5, and water deeper than 1.5B + D_f gives 1.0 and 1.0 — and the expressions above are the linear interpolation between them.

Comparing the four published factor sets

The table below lists the factors each method returns at a friction angle of 30°, together with the published source for the numbers. Every value in this table is reproduced exactly by the calculator, which is the cheapest available check that the implementation is faithful.

Notice that Terzaghi is the outlier on N_c and N_q and that the three Prandtl–Reissner methods differ only in the unit weight term, where Vesic is roughly 40 % above Meyerhof and Hansen. Because the unit weight term scales with footing width, the choice of N_γ matters most for wide footings on cohesionless soil and barely at all for narrow footings on clay.

Worked example: a square pier footing on silty sand

Consider a 2.5 m square pier footing founded 1.5 m below finished grade in a medium dense silty sand with c = 5 kPa, φ = 32° and γ = 19 kN/m³. The ground water table sits 4.0 m below grade. Use the FHWA/AASHTO factor set, general shear, and a factor of safety of 3.0.

1. Bearing capacity factors. N_q = e^{π tan 32°} tan²(61°) = 23.18; N_c = (23.18 − 1) cot 32° = 35.49; N_γ = 2(23.18 + 1) tan 32° = 30.21.
2. Shape factors with r = B/L = 1: s_c = 1 + 23.18/35.49 = 1.653; s_q = 1 + tan 32° = 1.625; s_γ = 1 − 0.4 = 0.600.
3. Ground water factors. D_w = 4.0 m is below founding level, so C_Wq = 1.0. For the unit weight term, 1.5B = 3.75 m, so C_Wγ = 0.5 + 0.5(4.0 − 1.5)/3.75 = 0.833.
4. Surcharge. q = 19 × 1.5 = 28.5 kPa.
5. Cohesion term. 5 × 35.49 × 1.653 = 293.3 kPa.
6. Surcharge term. 28.5 × 23.18 × 1.0 × 1.625 = 1073.3 kPa.
7. Unit weight term. 0.5 × 19 × 2.5 × 30.21 × 0.833 × 0.600 = 358.8 kPa.
8. Totals. q_ult = 293.3 + 1073.3 + 358.8 = 1725.4 kPa; net ultimate = 1725.4 − 28.5 = 1696.9 kPa; q_all = 1725.4/3.0 = 575.1 kPa; net allowable = 1696.9/3.0 = 565.6 kPa.

Pressing Reset to worked example loads exactly these inputs, so you can confirm the tool reproduces the arithmetic above before you trust it on your own numbers.

Interpreting the result and typical soil parameters

The number to hand to the structural engineer is normally the net allowable pressure, because the structural model usually already carries the weight of the backfill over the footing. If you quote the gross allowable pressure instead, say so explicitly in the foundation report — Circular No. 6 is emphatic that mixing the two silently double-counts the surcharge.

A result in the region of 500 to 900 kPa on a competent granular soil is normal and is almost never the value that sizes the footing. Serviceability governs: a footing loaded to several hundred kPa on sand will usually settle more than the superstructure tolerates long before shear failure is approached. Use the bearing capacity result as a ceiling, then size the footing on settlement.

Any apparent cohesion entered for a clean sand should be treated with suspicion. The cohesion term is multiplied by N_c, which is the largest of the three factors at low friction angles, so a spurious 10 kPa of cohesion can inflate the computed capacity by several hundred kPa.

Assumptions and limitations of this bearing capacity model

• One homogeneous layer. The soil is assumed uniform and isotropic to a depth of at least 1.5 to 2 times the footing width. Layered profiles, soft layers beneath a crust and footings on rock all need different methods.
• Vertical, concentric load on a level base. The base inclination factors b and the load inclination factors i of Equation 5-14 are set to 1.0. For eccentric loading, reduce the plan dimensions to the effective dimensions B′ and L′ before entering them, as Circular No. 6 directs, and check sliding separately.
• Embedment depth factor omitted. The factor d_q of Table 5-4 is taken as 1.0. Circular No. 6 explicitly permits this as a conservative simplification and requires it whenever the material above founding level is weaker than the bearing stratum.
• No sloping ground. Footings on or near a slope require the modified factors N_cq and N_γq and a separate global stability analysis.
• No settlement, no structural design. This is a shear failure check only. Immediate settlement, consolidation settlement, differential movement, footing thickness and reinforcement are all out of scope.
• Service load design, not LRFD. The factor of safety used here is an allowable stress design factor. It is not the reciprocal of an AASHTO resistance factor and the two must not be mixed.
• Consistent SI units only. Stresses in kPa, unit weight in kN/m³, lengths in metres, angles in degrees.

The output is a preliminary and educational estimate. Final foundation design must be carried out or reviewed by a qualified geotechnical engineer with access to site-specific investigation data.

Sources. Equations, correction factors and tabulated bearing capacity factors on this page are taken from: Federal Highway Administration, Geotechnical Engineering Circular No. 6 – Shallow Foundations, FHWA-SA-02-054, 2002 — Equations 5-2, 5-3, 5-4, 5-8, 5-9, 5-10, 5-13 and 5-14, Tables 5-1 to 5-4, and Section 5.2.6 (fhwa.dot.gov); US Army Corps of Engineers, Engineering and Design – Bearing Capacity of Soils, EM 1110-1-1905, 1992 — Tables 4-1, 4-2 and 4-4 for the Terzaghi, Meyerhof, Hansen and Vesic factor sets (publications.usace.army.mil); and the AASHTO LRFD Bridge Design Specifications, Article 10.6.3.1.2, which adopts the same general bearing capacity equation and factor set as FHWA. Original theory: Prandtl (1920), Reissner (1924), Caquot and Kerisel (1948), Terzaghi (1943), Meyerhof (1963), Brinch Hansen (1970) and Vesic (1973).