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: Frame the depth criterion as a single equation in $z$.
The exploration should stop where the load-induced stress drops to a tenth of the natural overburden stress: $\sigma_z = 0.1\,\sigma_v'$. Both sides depend on $z$, so this condition can be combined into one cubic equation and solved directly for $z$, without treating the two stresses separately at every step.

Step 2: Combine the two stress expressions in one line.
$\sigma_z = \dfrac{3P}{2\pi z^2}$ (Boussinesq stress on the load axis) and $\sigma_v' = \gamma' z$ with $\gamma' = \gamma_{sat} - \gamma_w = 18 - 10 = 8$ kN/m$^3$ (water table at the surface makes the soil submerged throughout). Setting $\sigma_z = 0.1\sigma_v'$:
\[ \frac{3P}{2\pi z^2} = 0.1\gamma' z \implies z^3 = \frac{3P}{0.2\pi\gamma'} \]

Step 3: Plug in numbers.
$P = 2000$ kN, $\gamma' = 8$ kN/m$^3$
\[ z^3 = \frac{3(2000)}{0.2\pi(8)} = \frac{6000}{1.6\pi} = \frac{6000}{5.0265} = 1193.66 \]

Step 4: Take the cube root.
$z = 1193.66^{1/3}$. Since $10.60^3 = 1191.0$ and $10.61^3 = 1194.4$, interpolating between the two gives $z \approx 10.61$ m.

Step 5: Sanity-check the order of magnitude.
A depth of about 10.6 m for a 2000 kN column load is a reasonable exploration depth in practice, since the zone of significant stress increase for such a load typically extends to one and a half to two times the loaded width or more, so the result is physically sensible.
\[ \boxed{z \approx 10.61 \text{ m}} \]