Trench bottom heave safety factor

Arching effect

Arching is the transfer of pressure from a yielding part of a soil mass to the adjacent stationary parts through the shear resistance of the soil. The pressure on the yielding part decreases and that on the stationary part increases (Terzaghi's trap door experiment).

Factor of safety against bottom heave (Terzaghi)

Rapid excavation means undrained conditions, so $\phi_u = 0$ and $c = c_u = 35$ kN/m$^2$. The heave is checked as a bearing failure of the base under the weight of soil beside the cut, resisted by shear on vertical planes. The bearing capacity of the base is taken as $5.7c$.

(a) Deep clay (no hard layer; $T \ge 0.7B$): the width of the failing strip is $B' = B/\sqrt2 = 0.7B$:

$$FS = \frac{5.7c}{\gamma H - \sqrt{2}\,cH/B} = \frac{5.7\times35}{18\times10 - \sqrt2\times35\times10/5} = \frac{199.5}{81.01} = 2.46$$

(b) Hard rock 2.5 m below the base: here $T = 2.5\ \text{m} < 0.7B = 3.5$ m, so the failure zone is limited by the rock and $B' = T = 2.5$ m:

$$FS = \frac{5.7c}{\gamma H - cH/T} = \frac{199.5}{180 - 35\times10/2.5} = \frac{199.5}{40.0} = 4.99$$

Answer: $FS = 2.46$ for deep clay. This is above the usual minimum of about 1.5, so the base is safe against heave, although the margin is modest. With the rock 2.5 m below the trench bottom, $FS = 4.99$, which is more than adequate.