Drainage-Ditch Slope Stability, φᶜ ≈ 0: 18-Env-A3 Geotechnical and Hydrogeological Engineering, May 2015, Q3

Question 3: Drainage-Ditch Slope Stability, φᶜ ≈ 0 (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Given data
Quantity	Symbol	Value
Ditch depth	$H$	2.0 m
Side slope	—	2.5 horizontal : 1 vertical
Saturated unit weight	$\gamma_{sat}$	21 kN/m³
Undrained cohesion	$c_u$	30 kN/m² (kPa)

Find. (a) factor of safety $FS$ of the left side slope against undrained ($\phi_u\approx0$) shear failure; (b) qualitative effect of tile drainage pipes on stability.

Fig. Q3 — 2 m deep ditch, 2.5H:1V side slopes, with a trial toe circle sketched for the φᶜ ≈ 0 (undrained) stability check.

Approach. For a purely cohesive ($\phi_u=0$) slope, Taylor's stability-number chart gives $FS=N_0\,c_u/(\gamma H)$ directly from the slope angle $\beta$; since no firm stratum is reported, use the deep-homogeneous-clay ($D\to\infty$) branch of the chart and note the more conservative toe-circle bound as a check.

Slope angle and Taylor's stability number. $\beta=\arctan(1/2.5)=21.8^\circ$. Reading Taylor's $\phi_u=0$ chart (Das Ch. 12; Craig & Knappett Ch. 9) at this angle, with no firm layer indicated (homogeneous clay to depth), gives a stability number $N_0\approx6.45$.
Factor of safety. $$FS=\frac{N_0\,c_u}{\gamma_{sat}H}=\frac{6.45\times30}{21\times2}=\boxed{FS\approx4.6}.$$

Check: the depth $D$ to a firm stratum is not stated. $N_0\approx6.45$ assumes deep, uniform clay ($D\to\infty$); if a firm layer instead sits right at the toe elevation ($D=1$), the toe-circle-only chart value is $N_0\approx5.53$ for any $\beta<53^\circ$, giving $FS=5.53\times30/42=3.95$. Either assumption clears $FS=3$ comfortably — the slope is stable with a substantial margin for this shallow 2 m cut.

(b) Tile drains lower the pore-water pressure in the clay behind the slope face by intercepting seepage before it can build up toward the toe, which raises the (drained) effective stress and, over the long term, improves stability relative to the undrained analysis above. In the short term, however, the trenching itself locally disturbs and loosens the soil along the slope, and a pipe that discharges onto the slope face rather than being carried well away can locally saturate and soften the toe, which would reduce — not improve — the factor of safety there; proper outletting of the drains away from the slope face is what makes them a net stabilizing measure.

Quantity	Value
(a) Slope angle, $\beta$	21.8° (2.5H:1V)
(a) Factor of safety, $FS$	≈ 4.6 (≥ 3.9 even on the conservative toe-circle bound)
(b) Net effect of properly-outletted tile drains	Stabilizing (lowers pore pressure)

Concept & Theory

Topic. Taylor's stability-number method for $\phi_u=0$ (undrained, total-stress) slope stability.

Key relations. $FS=\dfrac{N_0\,c_u}{\gamma H}$; $\beta=\arctan(1/m)$ for an $m\text{H}:1\text{V}$ slope; for $\beta\ge53^\circ$ the critical circle is always a toe circle, while for $\beta<53^\circ$ the critical circle can be a toe, slope or (given a deep firm layer) base circle, and $N_0$ depends on the depth factor $D$.

Why this works. Taylor's charts are the closed-form solution of the limit-equilibrium slip-circle problem for the special case $\phi=0$, where the strength envelope is a horizontal line and the critical circle geometry depends only on $\beta$ (and $D$) — no iterative circle search is required, unlike the general $c$-$\phi$ case.

Common pitfall. Reading $N_0$ off the $\beta\ge53^\circ$ toe-circle curve for a flatter slope, or forgetting that for $\beta<53^\circ$ the answer is sensitive to an unstated depth-to-firm-layer assumption — state it explicitly rather than silently picking one curve.

Source. Das, Principles of Geotechnical Engineering (9th ed.), Ch. 12 — Slope Stability (Taylor's Method); Craig & Knappett, Craig's Soil Mechanics (8th ed.), Ch. 9.

Question 3 of 6: Drainage-Ditch Slope Stability, φᶜ ≈ 0

Question 3: Drainage-Ditch Slope Stability, φᶜ ≈ 0 (20 marks)