Application of Terzaghi's Equation to Different Footing Shapes and Water Table Effects

TL;DR

Terzaghi's bearing capacity equation helps estimate how much load a footing can safely support, but you need to adjust it for different footing shapes and the presence of groundwater. Footing shape factors modify the equation for strip, square, and circular footings, while water table effects reduce the soil's effective strength. Understanding these adjustments is crucial for accurate foundation design.

1. The Mental Model

Imagine a foundation pushing into the ground. The soil resists this push, but how much it resists depends on the shape of the foundation and if the soil is wet or dry. Terzaghi's equation gives us a way to calculate this resistance by accounting for these factors.

2. The Core Material

Terzaghi's ultimate bearing capacity ($q_u$) equation for a strip footing (the basic form) is:

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

Where:
* $c$ = cohesion of the soil
* $q = \gamma D_f$ = surcharge pressure at footing level ($D_f$ is depth of footing, $\gamma$ is unit weight of soil above footing)
* $\gamma$ = unit weight of soil below footing
* $B$ = width of the footing
* $N_c, N_q, N_\gamma$ = Terzaghi's bearing capacity factors (dimensionless, depend on friction angle $\phi$)

Adjusting for Footing Shapes

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The original equation is for a long strip footing. For other common shapes, we introduce shape factors that modify the $c$, $q$, and $\gamma B$ terms.

Here's how the ultimate bearing capacity ($q_u$) changes for different shapes:

• Strip Footing:
  $q_u = c N_c + q N_q + 0.5 \gamma B N_\gamma$ (No shape factors needed, as it's the base case)

• Square Footing:
  $q_u = 1.3 c N_c + q N_q + 0.4 \gamma B N_\gamma$
  (Notice the 1.3 for $c N_c$ and 0.4 for $\gamma B N_\gamma$)

• Circular Footing:
  $q_u = 1.3 c N_c + q N_q + 0.3 \gamma B N_\gamma$
  (Again, 1.3 for $c N_c$, but 0.3 for $\gamma B N_\gamma$. For circular footings, B is the diameter.)

Considering Water Table Effects

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The presence of a high water table significantly reduces the effective strength of the soil, which in turn reduces its bearing capacity. We account for this by adjusting the effective unit weight of the soil ($\gamma$) in the $q$ and $0.5 \gamma B N_\gamma$ terms, or by using reduction factors.

The most common approach is to use an effective unit weight ($\gamma'$) for submerged soil, where $\gamma' = \gamma_{sat} - \gamma_w$. $\gamma_{sat}$ is the saturated unit weight and $\gamma_w$ is the unit weight of water ($9.81 \text{ kN/m}^3$ or $62.4 \text{ lb/ft}^3$).

Here's how the water table position affects the equation:

graph TD
    A["Water Table Location Relative to Footing"] --> B["Case 1: Water Table above footing base (D_w < D_f)"]
    A --> C["Case 2: Water Table at footing base (D_w = D_f)"]
    A --> D["Case 3: Water Table below footing base (D_w > D_f)"]

    B --> B1["Replace γ in 'q' term with γ_avg: (D_w γ' + (D_f - D_w) γ_sat) / D_f"]
    B --> B2["Replace γ in '0.5 γ B Nγ' term with γ' (submerged unit weight)"]
    B1 -- Simplified as --> B3["Or use effective unit weights directly: q = (D_w * γ_eff_above_WT) + (D_f - D_w) * γ'_below_WT"]
    B2 --> B4["γ in 0.5 γ B Nγ becomes γ'"]

    C --> C1["No change to γ in 'q' term (still γD_f)"]
    C --> C2["Replace γ in '0.5 γ B Nγ' term with γ'"]

    D --> D1["If (D_f < D_w < D_f + B): Use averaged γ_effective in '0.5 γ B Nγ' term: γ_avg = γ' + (D_w - D_f)/B * (γ_dry - γ')"]
    D1 --> D2["γ in 'q' term is unchanged (γ D_f)"]
    D --> D3["If (D_w >= D_f + B): No water table effect; use original γ in both terms"]

Explanation of Water Table Cases:

• Case 1: Water table is above the footing base ($D_w < D_f$)

  • The surcharge term, $q = \gamma D_f$, needs adjustment. The soil above the water table uses its natural unit weight ($\gamma$), while the submerged portion uses its effective unit weight ($\gamma'$). A common simplification is to use an average unit weight for the $D_f$ depth, or more precisely, calculate $q$ as the sum of stresses from different layers.
  • For the $0.5 \gamma B N_\gamma$ term (representing soil below the footing), you use the effective unit weight, $\gamma'$, because the soil is fully submerged.

• Case 2: Water table is at the footing base ($D_w = D_f$)

  • The surcharge term, $q = \gamma D_f$, remains unchanged because all soil above the footing is above the water table.
  • For the $0.5 \gamma B N_\gamma$ term, you use $\gamma'$ because the soil immediately below the footing is saturated.

• Case 3: Water table is below the footing base ($D_w > D_f$)

  • If the water table is within a distance $B$ below the footing base ($D_f < D_w < D_f + B$), the bearing capacity is partially reduced. You use a weighted average effective unit weight for the $\gamma$ in the $0.5 \gamma B N_\gamma$ term.
    • $\gamma_{avg} = \gamma' + (D_w - D_f)/B \times (\gamma - \gamma')$
    • Where $\gamma$ is the unit weight of soil above the water table and $\gamma'$ is the effective unit weight below.
  • If the water table is at a depth equal to or greater than $D_f + B$ below the footing, it's considered to have no effect on bearing capacity, and you use the soil's natural unit weight ($\gamma$) for all terms.

Remember, the goal of these adjustments is to use the effective stress of the soil, as pore water pressure doesn't contribute to soil strength.

3. Worked Example

Let's calculate the ultimate bearing capacity for a square footing.

Given:
* Footing: Square, $B = 2.0 \text{ m}$
* Depth of footing: $D_f = 1.0 \text{ m}$
* Soil properties:
* Cohesion, $c = 10 \text{ kPa}$
* Friction angle, $\phi = 25^\circ$
* Unit weight of soil (above water table), $\gamma = 18 \text{ kN/m}^3$
* Saturated unit weight of soil, $\gamma_{sat} = 20 \text{ kN/m}^3$
* Unit weight of water, $\gamma_w = 9.81 \text{ kN/m}^3$
* Terzaghi's bearing capacity factors for $\phi = 25^\circ$:
* $N_c = 25.1$
* $N_q = 12.7$
* $N_\gamma = 9.7$

Scenario: Water Table at Footing Base ($D_w = D_f = 1.0 \text{ m}$)

1. Calculate Effective Unit Weight for Submerged Soil:
   $\gamma' = \gamma_{sat} - \gamma_w = 20 \text{ kN/m}^3 - 9.81 \text{ kN/m}^3 = 10.19 \text{ kN/m}^3$

2. Determine Surcharge Term ($q$):
   The soil above the footing is above the water table, so use $\gamma$.
   $q = \gamma D_f = 18 \text{ kN/m}^3 \times 1.0 \text{ m} = 18 \text{ kPa}$

3. Apply Square Footing Equation with Water Table Adjustment:
   For a square footing with water table at base:
   $q_u = 1.3 c N_c + q N_q + 0.4 \gamma' B N_\gamma$ (Note: $\gamma'$ used for the last term)

   Plug in the values:
   $q_u = (1.3 \times 10 \text{ kPa} \times 25.1) + (18 \text{ kPa} \times 12.7) + (0.4 \times 10.19 \text{ kN/m}^3 \times 2.0 \text{ m} \times 9.7)$
   $q_u = (326.3) + (228.6) + (79.16)$
   $q_u = 634.06 \text{ kPa}$

So, the ultimate bearing capacity for this square footing with the water table at its base is approximately $634 \text{ kPa}$.

4. Key Takeaways

• Footing shape matters: Different shape factors are applied to Terzaghi's equation for strip, square, and circular footings.
• Square and circular footings are more efficient: They distribute the load better than a strip footing, leading to higher bearing capacities for the same soil.
• Water table is a major factor: A high water table significantly reduces the soil's effective unit weight, thereby lowering its bearing capacity.
• Use effective unit weight: For submerged portions of soil, always use the effective unit weight (