Allowable Bearing Capacity and Failure Modes

TL;DR

Allowable bearing capacity is the maximum load a foundation can safely carry, considering both ultimate failure and settlement. Understanding different failure modes helps us predict how a foundation might fail under excessive load. Terzaghi's equation is a key tool for calculating ultimate bearing capacity, which then gets reduced by a factor of safety.

1. The Mental Model

Imagine a building sitting on the ground. The ground can only support so much weight before it either collapses or sinks too much. We need to figure out what that safe weight limit is, and also how the ground might fail if we exceed it.

2. The Core Material

When we design a foundation, we're really trying to make sure it won't fail or settle excessively under the building's weight. The allowable bearing capacity is the maximum pressure a foundation can apply to the soil without causing shear failure or unacceptable settlement. It's essentially the "safe" load.

To get to the allowable bearing capacity, we first calculate the ultimate bearing capacity ($q_u$), which is the absolute maximum pressure the soil can withstand before failing. This is where Terzaghi's equation comes in. Once we have $q_u$, we divide it by a factor of safety (FS) to get the net ultimate bearing capacity ($q_{u,net}$) or allowable bearing capacity ($q_{all}$), depending on whether we consider the surcharge pressure. The factor of safety accounts for uncertainties in soil properties, loads, and design assumptions. Typically, FS ranges from 2.5 to 3.0 for shear failure.

Terzaghi's Ultimate Bearing Capacity Equation

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For a continuous footing (strip footing) on cohesive-frictional soil, Terzaghi's equation is:

$q_u = c N_c + q N_q + 0.5 \gamma B N_\gamma$

Where:
* $q_u$ = ultimate bearing capacity
* $c$ = cohesion of the soil
* $q$ = surcharge load at foundation level (usually $\gamma D_f$, where $\gamma$ is soil unit weight and $D_f$ is embedment depth)
* $\gamma$ = unit weight of the soil below the foundation
* $B$ = width of the footing
* $N_c, N_q, N_\gamma$ = Terzaghi's bearing capacity factors, which depend on the soil's friction angle ($\phi$)

For square footings, the equation is modified:
$q_u = 1.3 c N_c + q N_q + 0.4 \gamma B N_\gamma$

And for circular footings:
$q_u = 1.3 c N_c + q N_q + 0.3 \gamma B N_\gamma$

Failure Modes

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Foundations can fail in a few distinct ways depending on the soil type, density, and depth of the footing. Understanding these helps us anticipate potential issues.

graph TD
    A["Excessive Foundation Load"] --> B{Soil Type & Density?};

    B -->|Dense/Stiff Soil| C["General Shear Failure"];
    C --> C1["Sudden, Catastrophic Failure"];
    C --> C2["Well-defined Shear Surface"];
    C --> C3["Bulging of Soil Surface"];
    C --> C4["Tilting of Foundation (often)"];

    B -->|Medium Dense/Stiff Soil| D["Local Shear Failure"];
    D --> D1["Significant Settlement before Failure"];
    D --> D2["Slight Bulging (not pronounced)"];
    D --> D3["Shear Surface not Fully Developed"];
    D --> D4["No Sudden Failure"];

    B -->|Loose/Soft Soil| E["Punching Shear Failure"];
    E --> E1["Large Settlement (often without visible shear)"];
    E --> E2["No Bulging of Soil Surface"];
    E --> E3["Vertical Movement of Foundation"];
    E --> E4["Shear Zone confined to Foundation Edges"];

• General Shear Failure: This occurs in dense or stiff soils. A well-defined failure surface develops, extending to the ground surface. Failure is sudden and catastrophic, often accompanied by tilting of the foundation and significant bulging of the ground surface adjacent to the footing. This is the failure mode Terzaghi's equation primarily describes.
• Local Shear Failure: This happens in medium-dense or medium-stiff soils. There's considerable settlement before a distinct failure surface appears. The shear surface isn't fully developed, and there's only slight, if any, bulging of the ground surface. Failure isn't as sudden as general shear.
• Punching Shear Failure: This occurs in loose soils or soft clays, and also with deep foundations. The foundation essentially "punches" through the soil, moving vertically downwards. There's no clear shear surface extending to the ground, and little to no bulging of the adjacent soil. Settlement is typically large.

When calculating $q_u$ for local or punching shear, Terzaghi suggested using reduced shear strength parameters ($c'$ and $\phi'$) in his equation to account for the less brittle failure behavior. For local shear, he suggested $c' = (2/3)c$ and $\tan\phi' = (2/3)\tan\phi$.

Allowable Bearing Capacity

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After finding $q_u$, the net ultimate bearing capacity is $q_{u,net} = q_u - q$. This is the pressure the foundation can add to the soil beyond the existing overburden pressure.

The net allowable bearing capacity is:
$q_{all,net} = \frac{q_{u,net}}{\text{FS}} = \frac{q_u - q}{\text{FS}}$

Or, if you prefer to think about the total allowable pressure:
$q_{all} = \frac{q_u}{\text{FS}}$ (This value is often directly compared to the applied foundation pressure)

You'll always need to consider settlement separately. Even if the soil can handle the load without shear failure, the foundation might settle too much, causing structural damage.

3. Worked Example

Let's calculate the allowable bearing capacity for a square footing on a cohesive-frictional soil.

Given:
* Footing width ($B$) = 2 m
* Embedment depth ($D_f$) = 1.5 m
* Soil unit weight ($\gamma$) = 18 kN/m$^3$
* Cohesion ($c$) = 15 kN/m$^2$
* Friction angle ($\phi$) = 25 degrees
* Factor of Safety (FS) = 3.0

Bearing Capacity Factors for $\phi = 25^\circ$ (from Terzaghi's tables):
* $N_c = 25.1$
* $N_q = 12.7$
* $N_\gamma = 9.7$

Step 1: Calculate surcharge ($q$)
$q = \gamma D_f = 18 \text{ kN/m}^3 \times 1.5 \text{ m} = 27 \text{ kN/m}^2$

Step 2: Calculate ultimate bearing capacity ($q_u$) for a square footing
$q_u = 1.3 c N_c + q N_q + 0.4 \gamma B N_\gamma$
$q_u = (1.3 \times 15 \times 25.1) + (27 \times 12.7) + (0.4 \times 18 \times 2 \times 9.7)$
$q_u = 489.45 + 342.9 + 139.68$
$q_u = 972.03 \text{ kN/m}^2$

Step 3: Calculate net ultimate bearing capacity ($q_{u,net}$)
$q_{u,net} = q_u - q = 972.03 - 27 = 945.03 \text{ kN/m}^2$

Step 4: Calculate net allowable bearing capacity ($q_{all,net}$)
$q_{all,net} = \frac{q_{u,net}}{\text{FS}} = \frac{945.03}{3.0} = 315.01 \text{ kN/m}^2$

So, the foundation can safely apply a net pressure of approximately 315 kN/m$^2$ to the soil without shear failure, after accounting for the existing overburden. You'd then compare this to the actual net pressure from your building.

4. Key Takeaways

• Allowable bearing capacity is the safe maximum pressure a foundation can exert on the soil.
• It's derived from the ultimate bearing capacity, divided by a factor of safety.
• Terzaghi's equation helps calculate the ultimate bearing capacity for different footing shapes.
• General shear failure is sudden and catastrophic in dense soils; local shear is less sudden in medium soils; punching shear involves deep settlement in loose soils.
• The factor of safety addresses uncertainties in soil properties and loads, typically ranging from 2.5 to 3.0.
• Always consider settlement limits in addition to shear failure when designing foundations.

Common Mistakes to Avoid:
- Forgetting to apply the correct shape factors (1.3, 0.4, 0.3) for square/circular footings in Terzaghi's equation.
- Not accounting for the surcharge ($q$) when calculating the net ultimate or net allowable bearing capacity.
- Using bearing capacity factors ($N_c, N_q, N_\gamma$) that don't correspond to the correct friction angle ($\phi$) or theory (e.g., mixing Terzaghi's with Meyerhof's).
- Confusing ultimate bearing capacity with allowable bearing capacity; the factor of safety is critical.
- Ignoring settlement analysis; a foundation might be safe against shear failure but still settle excessively.

5. Now Try It

You're designing a continuous strip footing (B = 1.8 m, $D_f$ = 1.2 m) on a clayey sand with $\gamma = 19 \text{ kN/m}^3$, $c = 10 \text{ kN/m}^2$, and $\phi = 20^\circ$. Assume General Shear Failure is expected. Use a Factor of Safety of 2.8.

Bearing Capacity Factors for $\phi = 20^\circ$ (Terzaghi):
$N_c = 17.7$, $N_q = 7.4$, $N_\gamma = 5.0$

Calculate the net allowable bearing capacity ($q_{all,net}$).

What success looks like: A single numerical value in kN/m$^2$ for $q_{all,net}$.