Bearing Capacity of Soil: Equations, Values and Examples

The bearing capacity of soil is the largest pressure a foundation can put on the ground before the soil under it fails in shear. Its ultimate value, $q_u$, is calculated from the soil’s strength ($c'$ and $\phi'$), its unit weight and the footing’s width and depth, with Terzaghi’s equation or the general bearing capacity equation. Divided by a factor of safety, usually 3, it gives the allowable bearing capacity that footings are sized with. Settlement then decides how much of that pressure a design can use.

Every equation below is applied to one footing: a 2.00 m square base, 1.50 m deep in silty sand, with the water table 0.50 m under it. The worked example solves it by hand and compares the result with Rischio’s engine and with Terzaghi’s equation.

Ultimate, allowable and net bearing capacity

• The ultimate bearing capacity, $q_u$, is the average pressure under the base at which the soil fails in shear.
• The allowable bearing capacity is $q_{all} = q_u/FS$. For foundations the factor of safety $FS$ is usually 3: it covers the uncertainty in the soil’s strength and keeps the footing far from the large movements that come before failure.
• The net bearing capacity, $q_u - q$, takes off the effective overburden at the base, $q$ ($\gamma D_f$ in dry soil), which the ground already carried before the excavation. The net allowable bearing capacity is $(q_u - q)/FS$.

A geotechnical report can give its allowable pressure on either basis, and checking a footing against the wrong one is a common mistake; gross and net bearing pressure works through a check on each. Rischio compares gross values: $q_u$ against the largest pressure the load puts under the base, $q_{max}$, and asks for $FS = q_u/q_{max} \ge 3$. Its vertical load $V$ is everything that reaches the soil at the base, so it should include the footing’s own weight and the backfill over it.

How soil fails under a footing

Vesić described three ways the ground gives way under a footing. Which one happens depends on how dense or stiff the soil is and on how deep the footing sits:

• General shear failure is typical of dense sand and stiff clay. A continuous slip surface runs from under the footing up to the ground surface, the soil beside the footing heaves, and the load reaches a clear peak before the footing sinks and tilts.
• Local shear failure shows up in medium-dense sand and firm clay. The slip surfaces form under the footing without clearly reaching the surface, and the load keeps rising with large settlement, with no clear peak.
• Punching shear failure belongs to loose sand and soft clay, and to deep footings. The footing pushes straight down, shearing the soil along its edges, and the ground around it doesn’t heave.

The bearing capacity equations describe general shear. For local shear, Terzaghi reduced the soil’s strength to two thirds before applying them, $\bar{c}' = \tfrac{2}{3}c'$ and $\tan\bar{\phi}' = \tfrac{2}{3}\tan\phi'$, so a $\phi'$ of 30° becomes 21.1°. In soil loose or soft enough to punch, settlement usually limits the footing long before its bearing capacity does.

Under a strip footing, general shear forms three zones on each side. Zone I is a wedge under the base that moves down with the footing as if it were part of it. Zones II shear radially and are bounded below by a logarithmic spiral. Zones III are passive Rankine zones, pushed up and out by zones II. In Prandtl’s solution, where the $N_c$ and $N_q$ factors below come from, the wedge’s sides rise at $45° + \phi'/2$ and the passive zones’ lower sides at $45° - \phi'/2$. For $\phi' = 30°$ the mechanism reaches about $1.6B$ under the base and spreads $4.8B$ to each side of the centre. The soil above the base doesn’t shear in this model: all it contributes is its weight, the overburden $q = \gamma D_f$ at the level of the base.

Terzaghi’s bearing capacity equation

Terzaghi (1943) wrote the first complete bearing capacity equation, for a strip footing under a vertical, centred load failing in general shear:

Each term is one source of resistance: the cohesion $c'$ along the slip surfaces, the overburden $q$ that the passive zones have to lift, and the weight of the soil inside the failure zone, with $\gamma$ the unit weight under the base and $B$ the footing’s width. For square and circular footings he raised the cohesion term and lowered the weight term:

His factors $N_q$ and $N_c$ have a closed form:

$N_\gamma$ is read from tables, which differ a little. With $\phi' = 0$, $N_c = 5.7$, $N_q = 1$ and $N_\gamma = 0$; with $\phi' = 30°$, $N_c = 37.16$, $N_q = 22.46$ and the tables give $N_\gamma$ between 19 and 20.

Terzaghi’s equation is still the quickest hand check. It has no depth factor, so the soil above the base counts only as weight, and no inclination factor, so it doesn’t cover a footing pushed sideways. For the embedded footing of the example it gives 26% less than the general equation.

The general bearing capacity equation

Meyerhof (1963) extended Terzaghi’s three terms with factors for the footing’s shape, its depth and the inclination of the load. This general equation is the one Rischio’s engine applies, in the studio and in the free bearing capacity calculator:

The bearing capacity factors are Reissner’s, Prandtl’s and Vesić’s:

De Beer’s shape factors use the effective dimensions $B'$ and $L'$, and are all 1 for a strip:

Hansen’s depth factors use the full width $B$, for $D_f/B \le 1$:

For a deeper base, $\tan^{-1}(D_f/B)$, in radians, takes the place of $D_f/B$. The inclination factors, Meyerhof’s and Hanna and Meyerhof’s, depend on the inclination of the load, $\beta = \tan^{-1}(H/V)$, and $F_{\gamma i}$ drops to 0 once $\beta \ge \phi'$:

With $\phi' = 0$, the undrained case of a saturated clay loaded faster than it can drain, the strength is the undrained shear strength $s_u$, $N_c = 5.14$, $N_q = 1$, $N_\gamma = 0$ and $F_{cd} = 1 + 0.4\,D_f/B$, so $q_u = 5.14\,s_u F_{cs} F_{cd} + q$. For a footing on clay, the end of construction is usually the critical moment: the clay gains strength as it drains afterwards.

The strength parameters $c'$ and $\phi'$ are those of the stratum the base rests on, so a weaker layer within the failure zone, which reaches about $1.6B$ under the base for $\phi' = 30°$, makes the result unsafe and calls for a layered analysis. The overburden $q$ and the unit weight $\gamma'$ are effective values, which is where the water table comes in. The bearing capacity theory in the docs has every factor with its conditions.

How the water table changes the bearing capacity

Terzaghi’s principle of effective stress, $\sigma' = \sigma - u$, says that the soil’s strength depends on the stress its grains carry, the total stress less the pore water pressure. Below the water table each metre of soil adds $\gamma_{sat} - \gamma_w$ of effective stress instead of $\gamma$, roughly half as much. The equation uses two of these stresses, and where the water table sits decides how much it lowers them:

• Water above the base, at a depth $D_w < D_f$: $q = \gamma D_w + (\gamma_{sat} - \gamma_w)(D_f - D_w)$, and the soil under the base weighs $\gamma' = \gamma_{sat} - \gamma_w$.
• Water within $B$ under the base: $q = \gamma D_f$, and $\gamma'$ is the average of $\gamma$ above the water and $\gamma_{sat} - \gamma_w$ below it over the depth $B$ under the base, each weighted by its thickness.
• Water deeper than $D_f + B$: no effect.

Rischio’s engine works the same way with any number of strata: $q$ is the effective vertical stress at the base, and $\gamma'$ the thickness-weighted average between $D_f$ and $D_f + B$. This is what it gives for the example’s footing with the water table at different depths:

From dry to flooded, $q_u$ drops by 35% and the factor of safety falls from 3.98 to 2.60. The level to design for is the highest the water can reach during the structure’s life, often above the one measured while drilling.

Eccentric and inclined loads

A moment $M$ moves the resultant of the load off centre by $e = M/V$. Meyerhof’s approach keeps only the part of the base centred on the load: the effective width is $B' = B - 2e$, and $L' = L - 2e_L$ when the load is also off centre along the length. The general equation uses $B'$ in its third term and $B'/L'$ in the shape factors, while the depth factors keep the full $B$.

The pressure under the base stays linear over the whole footing, from $q_{min}$ to $q_{max} = \frac{V}{BL}\left(1 \pm \frac{6e}{B}\right)$. While $e \le B/6$, inside the middle third, the whole base presses on the soil; past it, $q_{min}$ turns negative and an edge lifts off. Rischio checks $q_u$, worked out on $B'$, against $q_{max}$.

Put a moment $M = 150$ kN·m along the width of the example’s footing: $e = 150/1{,}500 = 0.10$ m and $B' = 1.80$ m, so $q_u$ drops to 1,362.1 kPa while $q_{max}$ rises to $375.0 \times 1.3 = 487.5$ kPa ($q_{min} = 262.5$ kPa). The factor of safety falls from 3.77 to 2.79, a quarter less and below 3, for a load moved 10 cm.

A horizontal load $H$ inclines the resultant by $\beta = \tan^{-1}(H/V)$. With $H$ a tenth of $V$, $\beta = 5.7°$, and for $\phi' = 30°$ the inclination factors are $F_{ci} = F_{qi} = 0.88$ and $F_{\gamma i} = 0.66$.

Worked example: a square footing over the water table

A column brings 1,500 kN to the base of a square footing 2.00 m wide, founded 1.50 m deep in a silty sand. The water table is 2.00 m deep, 0.50 m under the base. Take $\gamma_w = 9.81$ kN/m³.

Step 1: bearing capacity factors

Step 2: overburden and unit weight under the base

The water table is under the base, so the overburden is that of the dry soil, $q = \gamma D_f = 18 \times 1.50 = 27.0$ kPa. The 2.00 m under the base, from 1.50 to 3.50 m deep, hold 0.50 m above the water table and 1.50 m below it, where the soil weighs $20 - 9.81 = 10.19$ kN/m³:

Step 3: shape, depth and inclination factors

The load is centred, so $B' = L' = 2.00$ m and $B'/L' = 1$:

With $D_f/B = 1.50/2.00 = 0.75$:

The load is vertical, so the three inclination factors are 1.

Step 4: ultimate bearing capacity

Term by term, carrying every decimal:

Step 5: allowable bearing capacity and factor of safety

The footing passes, with an allowable bearing capacity of 471.6 kPa (about 9,850 psf). Rischio’s engine gives the same: $q_u = 1{,}414.8$ kPa and $FS = 3.77$, with $q = 27.0$ kPa, $\gamma' = 12.14$ kN/m³ and every factor above in its calculation report.

Terzaghi’s value beside it

Terzaghi’s equation for a square footing, with his factors at 30° and the same $q$ and $\gamma'$, gives a lower value. Its $N_\gamma = 20.12$ comes from the closed-form fit to his tabulated values that the calculator uses:

That is 26% less than the general equation, and a factor of safety of $1{,}043.3/375.0 = 2.78$, so by Terzaghi alone the footing would have to grow. Most of the gap is the overburden term, 606.3 against 953.3 kPa. Terzaghi’s equation has no depth factor and no shape factor on that term, while the general equation credits the 1.50 m of soil over the base with shear resistance ($F_{qd} = 1.217$) and raises the term for a square base ($F_{qs} = 1.577$).

The calculator below opens on this same footing without the water. Switch the water table on, at 2 m, and it gives the $q_u$, the factor of safety and Terzaghi’s value worked out above.

To keep the model, try other strata and loads or get a calculation report with every step, build the footing in Rischio’s footing tool.

Settlement often governs before bearing capacity

Give the example’s silty sand an elastic modulus of 15 MPa and a Poisson’s ratio of 0.3, and Rischio’s footing tool estimates an elastic settlement of 32.3 mm at the centre of the base, more than the 25 mm it allows by default. The footing that carries its load with $FS = 3.77$ fails on settlement. A 3.00 m square footing brings the settlement down to 23.9 mm, with a factor of safety of 8.45 on bearing. On sand, and under wide footings, settlement often governs like this.

The two checks change differently with the footing’s size. Under the same pressure, a wider footing has more bearing capacity, through the $\tfrac{1}{2}\gamma' B' N_\gamma$ term, and also settles more, because the elastic settlement grows in proportion to the width. Past a certain width, settlement sets the allowable pressure.

In clays the settlement that matters is usually consolidation, which is slow and often larger than the elastic part. Rischio’s elastic estimate leaves it out, and a warning says so with every settlement result. How foundation settlement is calculated covers both, and the settlement theory in the docs has the method the footing tool uses.

How the bearing capacity is found on a real project

Every number in the equations comes from the geotechnical investigation: borings, field tests and laboratory tests on the samples, with the water level read in the boreholes. How to read a geotechnical report follows that chain from a boring log to an allowable pressure.

Laboratory strength tests

The direct shear test (ASTM D3080) gives the drained strength parameters $c'$ and $\phi'$, mostly of sands, and the consolidated-undrained triaxial test with pore pressure readings (ASTM D4767) gives them for clays. The unconsolidated-undrained triaxial test (ASTM D2850) and the unconfined compression test (ASTM D2166) give the undrained strength $s_u$ for the short-term case. The Academy’s direct shear lesson and the direct shear calculator go from a test’s readings to $c'$ and $\phi'$.

SPT and CPT correlations

The standard penetration test (ASTM D1586) counts the hammer blows that drive a split-spoon sampler 300 mm into the soil, after the first 150 mm of seating. Corrected to 60% of the hammer’s energy, the blow count $N_{60}$ correlates with $\phi'$ in sands, and Meyerhof’s correlations turn it straight into the allowable pressure for 25 mm of settlement. The cone penetration test (ASTM D5778) gives a continuous profile of cone resistance, which correlates with $\phi'$ in sands and gives the undrained strength of clays as $s_u = (q_t - \sigma_{v0})/N_{kt}$, with $N_{kt}$ usually between 10 and 20.

Plate load tests

A rigid steel plate, typically 0.3 to 0.76 m across, is loaded in steps at the level of the future base while its settlement is measured. In clay, the footing’s ultimate bearing capacity is taken as the plate’s; in sand, it grows in proportion to the width. The plate only stresses the soil to a depth of about twice its width, so a soft layer at the depth a real footing reaches goes unseen.

Typical values: the IBC’s presumptive load-bearing values

Without a geotechnical study, the International Building Code allows the presumptive load-bearing values of its Table 1806.2, allowable pressures by class of material:

Higher values need data to back them, and mud, organic silt, organic clay, peat and unprepared fill get no presumptive value at all. The soil classes use the symbols of the Unified Soil Classification System; the USCS classification calculator puts a soil in its group from its gradation and its Atterberg limits.

These are conservative values for a soil nobody has tested, so use them as an order of magnitude. The example’s silty sand, an SM, would get 2,000 psf (96 kPa), against the 471.6 kPa its calculation allows, and the settlement check still has the last word on that.

Which soils have the best bearing capacity

Rock carries the most, then dense gravel, dense sand and stiff clay. Loose sand, soft clay and silt carry little, and organic soils, peat and uncontrolled fill aren’t fit to found on without treatment. The IBC’s table follows the same order, from 12,000 psf for crystalline bedrock down to 1,500 psf for clays and silts.

The ideal soil for a foundation carries the load on a small footing and settles little and evenly: dense, well-graded sand and gravel or stiff clay, uniform across the site and above the water table. Expansive clays, collapsible soils and loose saturated sands, which can liquefy in an earthquake, need more than a bearing check. When the good stratum lies too deep for a footing, piles take the load down to it, as in pile foundation design; when only the top few metres are poor, ground improvement can treat them in place.

Frequently asked questions

How do you know the bearing capacity of soil?

From a geotechnical investigation. Borings with SPT or CPT soundings and laboratory strength tests give $c'$ and $\phi'$, or $s_u$ for clays, and a bearing capacity equation turns them into $q_u$; a plate load test measures it at one spot. Where no study is required, the IBC’s presumptive values give a conservative allowable pressure by soil class.

Which soil has the best bearing capacity?

Rock, followed by dense gravel and sand and then stiff clay. In the IBC’s presumptive values, crystalline bedrock takes 12,000 psf (575 kPa), gravel 3,000 psf (144 kPa), sand 2,000 psf (96 kPa), and clay and silt 1,500 psf (72 kPa).

What is a good soil bearing capacity?

There’s no single figure: a soil is good for a foundation when its allowable pressure keeps the footings a practical size and the settlement within limits. As an order of magnitude, the IBC’s presumptive values for soils run from 1,500 psf (72 kPa) for clays and silts to 3,000 psf (144 kPa) for gravels, and a geotechnical study can justify higher values for the actual soil and footing.

Which soil is ideal for a foundation?

Dense, well-graded gravel and sand or stiff clay, uniform across the site and above the water table. Organic soils, peat, uncontrolled fill, expansive clays and loose saturated sands are the ones to avoid or treat.

What units is bearing capacity measured in?

It’s a pressure: kPa (kN/m²) in SI units, psf, ksf or tsf in US units, and kg/cm² or t/m² in much of Latin America. 1 ksf is 47.9 kPa and 1 kg/cm² is 98.1 kPa.

Does the water table reduce bearing capacity?

Yes, once it rises to within a footing’s width under the base. Submerged soil weighs about half as much, which lowers the weight term, and a water table above the base lowers the overburden term too. For the example’s footing, a water table rising from deep down to the ground surface cuts $q_u$ by 35%.