Question

A building is proposed in an area having thick deposit of silty clay. The water table is at the ground surface. The saturated unit weight of soil is 18 kN/m³ and unit weight of water is 10 kN/m³. The maximum vertical load (\(P\)) on a column of the proposed building is 2000 kN.

Consider \(\sigma_z \leq 0.1\sigma_v'\) for computation of the minimum depth of soil exploration.

\(\sigma_v'\) is the effective vertical overburden stress. \(\sigma_z\) is the increase in the vertical stress at depth \(z\) below load \(P\) as per the Boussinesq's stress theory.

Based on above, the minimum depth (in m) of soil exploration required for the foundation design is (rounded off to two decimal places).

Hint:

Use Boussinesq's point load formula \(\sigma_z = 3P/(2\pi z^2)\) directly below the load, and the submerged unit weight for \(\sigma_v'\) since the water table is at the surface; solve \(\sigma_z = 0.1\sigma_v'\) for \(z\).

Answer 1

Correct Answer: 10.61

Step 1: Understand what governs the depth of exploration.
Soil exploration should go deep enough that the extra stress the new column load adds to the ground becomes small compared with the stress already carried by the soil's own weight. The stopping criterion given is \(\sigma_z \leq 0.1\sigma_v'\): explore until the induced stress \(\sigma_z\) falls to one tenth of the existing effective overburden stress \(\sigma_v'\) at that depth. The minimum depth is the value of \(z\) where equality holds.

Step 2: Write \(\sigma_z\) directly below the point load using Boussinesq's theory.
Directly under a point load \(P\) (radial distance \(r=0\)), Boussinesq's stress influence formula simplifies to
\[ \sigma_z = \frac{3P}{2\pi z^2} \]

Step 3: Write \(\sigma_v'\), the effective overburden stress.
The water table is at the ground surface, so the soil below is fully submerged all the way down. The effective (submerged) unit weight is
\[ \gamma' = \gamma_{sat} - \gamma_w = 18 - 10 = 8 \text{ kN/m}^3 \]
so at depth \(z\), \(\sigma_v' = \gamma' z = 8z\).

Step 4: Apply the stopping condition \(\sigma_z = 0.1\sigma_v'\).
\[ \frac{3P}{2\pi z^2} = 0.1(8z) \]
\[ \frac{3(2000)}{2\pi z^2} = 0.8z \]
\[ \frac{3000}{\pi z^2} = 0.8z \]

Step 5: Solve for \(z\).
\[ z^3 = \frac{3000}{0.8\pi} = \frac{3000}{2.5133} = 1193.66 \]
\[ z = (1193.66)^{1/3} \approx 10.61 \text{ m} \]

Final Answer:
The minimum depth of soil exploration required is
\[ \boxed{z \approx 10.61 \text{ m}} \]