Mat 10 m × 15 m with basement, FOS

Given: mat $B = 10$ m, $L = 15$ m, $D_f = 2.5$ m, saturated clay $c_u = 30$ kN/m², total load $Q = 4000$ kN. The unit weight of soil is not given.

Assumption: the relief due to excavation is ignored (conservative) because $\gamma$ is not given, so the net applied pressure is taken equal to $Q/A$.

Bearing capacity factor (Skempton)

$$\begin{aligned} N_{c(sq)} &= 6\left(1 + 0.2 \times \frac{2.5}{10}\right) = 6.30 \\ N_{c(rect)} &= \left(0.84 + 0.16\times\frac{10}{15}\right)(6.30) = 5.96 \\ q_{nu} &= c_uN_c = 30 \times 5.964 = 178.9\ \text{kN/m}^2 \end{aligned}$$

Applied pressure and factor of safety

$$\begin{aligned} q &= \frac{4000}{10 \times 15} = 26.67\ \text{kN/m}^2 \\ FS &= \frac{178.9}{26.67} = 6.71 \end{aligned}$$

Answer: factor of safety $\approx 6.7$. If the weight of the soil excavated for the basement ($\gamma D_f \approx 18 \times 2.5 = 45$ kN/m²) is taken into account, it exceeds the applied pressure of 26.7 kN/m², so the mat is over-compensated and the true factor of safety is even higher.