Designing a safe and efficient foundation is crucial for any structural project. In this article, we explore how to determine the foundation dimension using Terzaghi’s bearing capacity formula with a factor of safety of 3. We evaluate two scenarios: when the water table is one meter below the foundation base, and when it rises to the ground surface.

Understanding the Problem

A square footing must be designed to carry a load of 2400 kN. Soil parameters are:

• Cohesion (c) = 50 kN/m²
• Angle of internal friction (φ) = 32°
• Unit weight of soil above water table (γd) = 17.25 kN/m³
• Saturated unit weight (γsat) = 19.5 kN/m³

We use Terzaghi’s general shear failure formula:

qu = 1.3 * c * Nc + q * Nq + 0.4 * B * γ * Nγ

Where:

• qu = ultimate bearing capacity
• Nc, Nq, Nγ = bearing capacity factors depending on φ
• q = effective overburden pressure
• B = foundation width

Applied load (Qall) = 2400 kN
Factor of Safety (FS) = 3

Thus, required ultimate load capacity:
qu = (Qall / Area) * FS = (2400 / B²) * 3 = 7200 / B²

Case A: Water Table at 1 m Below Foundation Base

Effective unit weight γ’ is reduced due to partial submergence:

γ’ = γsat – γw = 19.5 – 10 = 9.5 kN/m³ , d = 3 -2 = 1m

Equivalent unit weight for calculations:

γ = γˉ =  γ’ ​+ d (γd – γ’)/B

γ = 9.5 + 1(17.25 – 9.25)/B

γ = 9.5 + (7.75 / B)

Overburden pressure:
q (effective stress) = γd * Df = 17.25 * 2 = 34.5 kN/m²

Bearing capacity factors for φ = 32° are:

• Nc = 44.04
• Nq = 28.52
• Nγ = 26.87

Substitute values:

qu = 1.3 * c * Nc + q * Nq + 0.4 * B * γ * Nγ

7200 / B² = 1.3 * 50 * 44.04 + 34.5 * 28.52 + 0.4 * B * (9.5 + 7.75/B) * 26.87

After simplification:
7200 / B² = 3923.837 + 102.106B

Solving the cubic equation gives:
B ≈ 1.33 m

Case B: Water Table at Ground Surface

Here, the effective stress reduces further:

q = (γsat – γw) * Df = (19.5 – 10) * 2 = 19 kN/m²

γ = γ’ = γsat – γw

          = 19.5 – 10 = 9.5 kN/m³

Using Terzaghi’s formula:

qu = 1.3 * c * Nc + q * Nq + 0.4 * B * γ * Nγ

7200 / B² = 1.3 * 50 * 44.04 + 19 * 28.52 + 0.4 * B * 9.5 * 26.87

Simplify:
7200 / B² = 3404.48 + 102.106B

Final solution:
B ≈ 1.42 m

Key Insights

• The water table level directly affects bearing capacity.
• When the water table rises to the surface, the required foundation width increases.
• Proper design ensures stability even with fluctuating groundwater conditions.
• A factor of safety of 3 provides additional security against shear failure.

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