Square footing under eccentric load, min depth

Given: $B = L = 4$ m, $\phi = 32^\circ$, $c = 35$ kN/m², $\gamma = 18$ kN/m³, $Q = 2000$ kN, $e = 0.3$ m in one direction, FS = 2.5, $N_c = 55$, $N_q = 40$, $N_\gamma = 45$.

Method: use the effective width $B' = B - 2e$ (Meyerhof) and Terzaghi's square footing equation $q_u = 1.3cN_c + \gamma D_f N_q + 0.4\gamma B' N_\gamma$. The load must satisfy $\dfrac{q_u}{FS} A' \ge Q$.

$$\begin{aligned} B' &= 4 - 2(0.3) = 3.4\ \text{m},\quad A' = 3.4 \times 4 = 13.6\ \text{m}^2 \\ \text{Required } q_u &= \frac{2.5 \times 2000}{13.6} = 367.6\ \text{kN/m}^2 \end{aligned}$$

Terms of $q_u$ (independent of depth):

$$\begin{aligned} 1.3cN_c &= 1.3 \times 35 \times 55 = 2502.5\ \text{kN/m}^2 \\ 0.4\gamma B' N_\gamma &= 0.4 \times 18 \times 3.4 \times 45 = 1101.6\ \text{kN/m}^2 \end{aligned}$$

So even at the ground surface ($D_f = 0$): $q_u = 2502.5 + 1101.6 = 3604.1\ \text{kN/m}^2$, which is far more than the required $367.6$ kN/m². The factor of safety at $D_f = 0$ is $3604.1 \times 13.6/2000 = 24.5$.

Answer: from the shear failure condition, no minimum depth is required (the requirement is met even at $D_f \approx 0$, FS = 24.5 against 2.5). The depth is therefore decided by other needs: a minimum of about 1.0 m below the ground (to get below topsoil, frost and shrinkage zones) is adopted in practice, which gives FS = 29.4.

Note: with the data given, the load is small compared with the capacity of the soil, so shear failure does not control the depth.