Coarse material thickness against piping

A layer of coarse material (a loaded filter) placed on top of the soil adds weight to resist the upward seepage force. The head loss in this material is negligible.

Given

Soil: $L = 1.15$ m, $n = 0.30$, $G = 2.7$. Upward seepage head at the base $h = 1.95$ m. Required factor of safety $F = 2$. Coarse layer of thickness $z$ with the same $G$ and $n$.

Submerged unit weight

$$e = \frac{n}{1-n} = \frac{0.30}{0.70} = 0.4286$$ $$\gamma' = \frac{(G-1)\gamma_w}{1+e} = \frac{1.7}{1.4286}\gamma_w = 1.19\,\gamma_w$$

Factor of safety against piping

At the base of the soil stratum, the effective stress with the load of the coarse layer:

$$\sigma' = \gamma'(L + z) - \gamma_w h$$

Factor of safety = submerged weight / uplift:

$$F = \frac{\gamma'(L+z)}{\gamma_w h} = 2$$ $$1.19\,(1.15 + z) = 2 \times 1.95 = 3.9$$ $$1.15 + z = 3.277 \;\Rightarrow\; z = 2.13\ \text{m}$$

For comparison, without the coarse layer $F = \dfrac{1.19\times1.15}{1.95} = 0.70 < 1$, so the stratum would pipe.

Answer: Thickness of coarse material required $\approx 2.13$ m.