GATE 2023 CE (CE1) – Question 58
A vertical trench is excavated in a clayey soil deposit having a surcharge load of 30 kPa. A fluid of unit weight 12 kN/m$^3$ is poured in the trench to prevent collapse as the excavation proceeds. Assume that the fluid is not seeping through the soil deposit. If the undrained cohesion of the clay deposit is 20 kPa and saturated unit weight is 18 kN/m$^3$, what is the maximum depth of unsupported excavation (in m, rounded off to two decimal places)?
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Correct answer: 3.31 to 3.35
Explanation
In a clay with $\phi_u = 0$, the active pressure at depth $z$ is $\gamma z + q - 2c = 18z + 30 - 40 = 18z - 10$ kPa. The force of the soil on the trench wall over the depth $h$ is $\int_0^h(18z - 10)dz = 9h^2 - 10h$. The fluid in the trench gives the resisting force $\frac{1}{2} \times 12 \times h^2 = 6h^2$. At the limit the two forces are equal: $9h^2 - 10h = 6h^2$, so $3h^2 = 10h$ and $h = 3.33$ m.