What is Terzaghi's bearing capacity formula?

Terzaghi (1943) proposed qu = c·Nc + q·Nq + 0.5·γ·B·Nγ for the ultimate bearing capacity of shallow foundations. Nc, Nq, and Nγ are dimensionless bearing capacity factors that depend on the soil friction angle φ. This formula is widely used in practice for preliminary foundation design.

How are Nc, Nq, and Nγ calculated?

Nq = exp(π·tanφ)·tan²(45+φ/2), Nc = (Nq−1)/tanφ (for φ > 0; Nc = 5.14 at φ = 0), Nγ = 2(Nq+1)tanφ. These factors grow rapidly with increasing friction angle, which is why even a few degrees of φ can dramatically increase bearing capacity.

How does footing shape affect bearing capacity?

Strip footings use shape factors of 1.0 (baseline). Square and circular footings use shape factors Fcs = 1 + Nq/Nc and Fqs = 1 + tanφ, which generally increase the cohesion and surcharge terms. The Nγ term is reduced by Fγs = 0.6. Overall, square and circular footings typically have higher bearing capacity than equivalent strip footings.

What safety factor should be used for allowable bearing capacity?

The allowable bearing capacity qa = qu / FS. A factor of safety FS = 2.5 to 3.0 is typically applied in practice (this tool defaults to FS = 3.0). Settlement analysis must also be conducted alongside bearing capacity checks for a complete foundation design.

› Soil Bearing Capacity Back

Geotechnical Engineering Tool

Soil Bearing Capacity Calculator

Compute ultimate and allowable soil bearing capacity using Terzaghi's formula. Adjust cohesion, friction angle, foundation geometry, and shape factor to instantly see how Nc, Nq, and Nγ influence your design.

Soil & Foundation Parameters

Cohesion c 20 kPa

Friction angle φ 30 °

Unit weight γ 18 kN/m³

Foundation width B 1.5 m

Embedment depth Df 1.0 m

Footing Shape

Safety factor FS 3.0

Bearing Capacity Factors

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N_c

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N_q

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N_γ

Results

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Ultimate q_u (kPa)

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Allowable q_a (kPa)

Terzaghi Bearing Capacity

$$q_u = cN_c F_{cs}+ qN_q F_{qs}+ \tfrac{1}{2}\gamma B N_\gamma F_{\gamma s}$$

where $q = \gamma D_f$ (surcharge)

Bearing capacity factors:

$$N_q = e^{\pi\tan\phi}\tan^2\!\left(45+\tfrac{\phi}{2}\right)$$ $$N_c = \frac{N_q-1}{\tan\phi},\quad N_\gamma = 2(N_q+1)\tan\phi$$

qu and qa vs friction angle φ (0–45°) using current c, γ, B, Df

qu and qa vs foundation width B (0.5–5 m) using current c, φ, γ, Df

What is Soil Bearing Capacity?

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What exactly is "soil bearing capacity"? Is it just how much weight the ground can hold?

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Basically, yes! It's the maximum average pressure the soil can withstand before it fails and the foundation sinks. In practice, we calculate an "ultimate" capacity ($q_u$), then divide by a safety factor to get the "allowable" pressure ($q_a$) we can safely design for. Try moving the "Cohesion" slider in the simulator above to see how a sticky clay soil can dramatically increase the calculated capacity.

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Wait, really? So the formula has three parts. What's the "surcharge" term ($qN_q$) for?

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Good observation! That term accounts for the soil *beside* the foundation. The "q" is the pressure from the soil column of depth $D_f$ around the footing. A common case is a basement wall footing—the deeper it's buried, the more the surrounding soil helps hold it up. In the simulator, increase the "Embedment Depth" ($D_f$) and watch the allowable capacity ($q_a$) rise, even if you don't change the soil's strength.

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So the "Friction Angle" must be super important too. What happens if I set it to zero, like for very soft clay?

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Exactly! For $\phi = 0$, the bearing capacity factors $N_q$ and $N_\gamma$ collapse to specific values. The capacity then depends almost entirely on cohesion ($c$). This models "undrained" clay conditions. Try it: set Friction Angle to 0 and Cohesion to, say, 50 kPa. Then, change the Foundation Width ($B$). You'll see the capacity barely changes, which is a key characteristic of clay versus sandy soil.

Physical Model & Key Equations

The ultimate bearing capacity $q_u$ is calculated using Terzaghi's general formula, which sums contributions from soil cohesion, surcharge from surrounding soil, and the soil's self-weight and friction.

$$q_u = cN_c F_{cs}+ qN_q F_{qs}+ \tfrac{1}{2}\gamma B N_\gamma F_{\gamma s}$$

$c$: Soil cohesion (kPa). $q$: Surcharge pressure = $\gamma D_f$. $\gamma$: Soil unit weight (kN/m³). $B$: Foundation width (m). $N_c, N_q, N_\gamma$: Bearing capacity factors. $F_{cs}, F_{qs}, F_{\gamma s}$: Shape factors (for square/rectangular footings).

The bearing capacity factors $N_q$ and $N_c$ are derived from plasticity theory and depend solely on the soil's internal friction angle $\phi$.

$$N_q = e^{\pi\tan\phi}\tan^2\!\left(45+\tfrac{\phi}{2}\right)$$ $$N_c = (N_q - 1)\cot\phi$$

These equations show the exponential relationship with $\phi$. A small increase in friction angle causes a large increase in $N_q$, which is why sandy soils ($\phi > 30°$) can have very high bearing capacity even with zero cohesion.

Real-World Applications

Residential Footing Design: Engineers use this calculation daily to size concrete footings for houses. For instance, on sandy soil, they might determine a 0.5m wide strip footing is sufficient, while on soft clay, they might need to recommend deep piles instead.

Industrial Slab Design: The heavy loads from factory machinery or warehouse storage racks must be supported by the ground floor slab. Calculating the bearing capacity ensures the slab won't crack or settle unevenly under point loads.

Bridge Abutment Design: The massive concrete supports at the ends of a bridge exert tremendous pressure on the ground. Geotechnical engineers perform this analysis, often with high safety factors, to ensure long-term stability.

Retaining Wall Stability Check: A key mode of failure for a retaining wall is it sliding forward or tilting due to the pressure of the soil behind it. The bearing capacity of the soil *under* the wall's base is a critical part of this check.

Common Misunderstandings and Points to Note

When using this tool for calculations, there are several pitfalls that beginners in particular often fall into. The first is the "selection of representative values for input parameters". For example, even if you input "internal friction angle φ=30°", sandy soil on-site is not uniform. In design, careful judgment is required, such as adopting the lower limit or average value from multiple test results. When experimenting with the tool, try comparing how the results change between "φ=25° and 35°" to get a feel for parameter sensitivity.

The second point is "converting calculation results to allowable bearing capacity". The "ultimate bearing capacity" output by this tool is the value at which the ground is on the verge of failure. In actual design, a safety factor FS (typically 3) is applied for safety, dividing the ultimate value to obtain the "allowable bearing capacity". If the ultimate bearing capacity is 300 kN/m², the actual allowable capacity is around 100 kN/m². It is extremely dangerous to forget this safety factor and use the ultimate value directly.

The third point is the "applicability limits of Terzaghi's formula". This formula assumes relatively shallow foundations (where the embedment depth Df is less than the foundation width B). Different theories are needed for deep foundations (like piles), sloping ground, or dynamic loads during earthquakes. Also, for clay with φ=0, Nγ=0, but this is for long-term stability calculations. A different approach is needed for short-term conditions (immediately after construction). Please understand that this tool is merely a "first step".

Related Engineering Fields

Calculating ground bearing capacity is not a standalone task; it is a crucial "piece" for comprehensively evaluating structural behavior. For example, in "retaining wall stability calculations", in addition to checking the wall itself for overturning and sliding, you must always verify that the ground directly beneath the wall does not fail due to insufficient bearing capacity (bearing capacity check). This is where the calculations from this tool come into play.

It is also deeply related to "predicting differential settlement". If the bearing capacity differs significantly under different parts of a building, or if the thickness of a soft layer is uneven, differences in settlement amount can occur. Bearing capacity calculation is the first step in judging whether the ground can support the load uniformly. In more advanced fields, there is "coupled soil-structure interaction analysis using the Finite Element Method (FEM)". The bearing capacity obtained with this tool serves as an important benchmark (comparison) value when constructing and validating complex FEM models.

In this way, starting from foundation design, through stability analysis of earth structures, and even to the verification of numerical simulations, the concept of ground bearing capacity acts as a vertical thread in geotechnical engineering, connecting various fields.

For Further Learning

Once you're comfortable with this tool and think "I want to know more", it's time to take the next step. First, "comparing it with other bearing capacity formulas" is recommended. The next most famous formula after Terzaghi's is Meyerhof's. Its characteristic is considering that the soil above the foundation base also contributes to shear resistance, often providing more realistic values, especially for deeply embedded foundations. Comparing the results of both clarifies the differences in theory.

Next, try delving deeper into the mathematical background. Why do the natural logarithm base *e* and pi (π) appear in the Nq formula? This is the result of mathematically expressing the failure mechanism known as the "logarithmic spiral slip surface". The derivation of the formula uses the limit analysis theory of plasticity mechanics. Following the equations can be challenging, but even just understanding the physical image behind them (the soil blocks sliding in a spiral shape) through diagrams will broaden your perspective.

Ultimately, the important step is expanding your focus from "bearing capacity" to "settlement". The ground settles under load even if it doesn't fail. Even if the allowable bearing capacity is sufficient, excessive settlement can cause cracks in the building. As your next learning topic, I strongly recommend tackling calculations for immediate settlement based on elastic theory and time-dependent settlement due to consolidation of clay layers. It is only when both bearing capacity and settlement considerations are in place that safe and economical foundation design is complete.