Question 9 of 9: Undrained stability of a canal bank cut in saturated clay
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Examinations, December 2013 — 98-Civ-B3 Geotechnical Design. Three hours, open book, any non-communicating calculator. Section A holds five 7-mark discussion questions (answer any four); Section B holds four 24-mark design questions (answer any three), so the examinable total is 4 × 7 + 3 × 24 = 100 marks. Every one of the nine questions is answered here, because the set is a study resource rather than a sitting.
Reference texts. B. M. Das, Principles of Foundation Engineering (9th ed.) and Principles of Geotechnical Engineering (9th ed.); Canadian Geotechnical Society, Canadian Foundation Engineering Manual (CFEM, 4th ed.) — the governing Canadian reference for foundation practice; R. F. Craig, Craig's Soil Mechanics (9th ed.); D. P. Coduto, Foundation Design: Principles and Practices (3rd ed.).
Sources of design charts and assumed values (paper Note 6). The paper requires every chart and assumed value to be identified. Bearing-capacity factors are Terzaghi’s (Das, Foundation Engineering, Table 3.1, with Nγ after Kumbhojkar 1993); the pile end-bearing factor Nq* is read from Meyerhof’s chart (Das Fig. 11.14) and cross-checked against Janbu’s closed-form expression; the adhesion factor α is Das Table 11.5 (Terzaghi, Peck & Mesri); the strain-influence distribution is Schmertmann, Hartman & Brown (1978). Assumed values — specific gravity of solids Gs = 2.65 for the Question 8 backfill, base friction δ = ⅔φ′ and base adhesion ca = ⅔c′ (CFEM §24), and a driving-parameter value K = 1.4K0 for a high-displacement driven pile (Das §11.11) — are flagged where they are used.
Check — Question 6 text and figure disagree. The printed text of Question 6 states a 1 m × 1 m square footing with γ = 20 kN/m³ and φ = 36°, while Figure 3 is drawn for a 2.5 m footing with γ = 18 kN/m³ and φ′ = 38°. The question text governs (it is the instruction to the candidate); the figure is used only for the embedment Df = 1.5 m and for the CPT modulus profile, which the text does not restate. The alternative reading is noted at the end of the answer.
Question 9: Undrained stability of a canal bank cut in saturated clay (24 marks)
$R = 15.75$ m, arc AED subtends $\theta = 95.5^{\circ}$
Area of the sliding mass ABCDE
$A = 155$ m$^2$ per metre run
Lever arm of the centroid G about O
$x = 3.2$ m
Clay (homogeneous, saturated)
$\gamma = 20$ kN/m$^3$, $c_u = 30$ kN/m$^2$
Tension zone CD
2.92 m
Bank slope
$30^{\circ}$
Find. The factor of safety of this trial slip surface with the canal empty, and how that value changes when the canal is filled to the top of the bank.
Figure Q9 — the trial circular slip surface AED, the sliding mass ABCDE and its centroid G. The disturbing moment is W acting at x = 3.2 m from the vertical through O; the restoring moment is cu acting along the whole arc at the radius R.
Approach. A freshly cut bank in saturated clay is loaded far faster than the clay can drain, so the critical condition is undrained and the analysis is the $\phi_u = 0$ Swedish circle method: the whole sliding mass is one free body rotating about O, the only resistance is the constant undrained strength mobilised along the arc, and the factor of safety is the ratio of the two moments about O.
Why the undrained analysis is the right one. Excavating the canal unloads the clay, so the pore pressures beneath the bank are initially negative and rise with time towards the steady-seepage values. The undrained strength therefore gives the highest factor of safety the slope will ever have, and the case examined here is the short-term one immediately after excavation. With $\phi_u = 0$ the strength is the constant $c_u = 30$ kPa and it acts everywhere along the arc, which is what makes the single-circle solution exact rather than approximate.
Weight of the sliding mass. Per metre run,$$W = A\gamma = 155(20) = 3100\ \text{kN/m}.$$
Disturbing moment about O. The weight acts vertically through the centroid G, whose horizontal offset from the vertical through O is $x = 3.2$ m:$$M_D = W x = 3100(3.2) = 9920\ \text{kN}\cdot\text{m/m}.$$
Length of the slip arc. The arc AED subtends $95.5^{\circ}$ at O, so$$L_a = R\theta = 15.75\left(95.5 \times \frac{\pi}{180}\right) = 15.75(1.6667) = 26.25\ \text{m}.$$The arc stops at D rather than continuing to C because the clay above D is in tension and cannot carry shear.
Restoring moment about O. The mobilised shear stress $c_u$ acts tangentially all along the arc, at the constant lever arm $R$:$$M_R = c_u L_a R = 30(26.25)(15.75) = 12\,404\ \text{kN}\cdot\text{m/m}.$$
Factor of safety with the canal empty.$$FS = \frac{M_R}{M_D} = \frac{12\,404}{9920}$$$$\boxed{FS = 1.25}$$This is adequate for a temporary condition but below the 1.4–1.5 usually required for a permanent slope, and the trial circle examined is only one of many — the critical circle would give a lower value still.
Check the tension crack. The depth over which the clay cannot sustain tension is$$z_0 = \frac{2c_u}{\gamma} = \frac{2(30)}{20} = 3.00\ \text{m},$$against the 2.92 m marked as CD on the figure — agreement to within 3 %, which confirms that the arc has been correctly terminated at D and that no extra allowance is needed. If the crack were to fill with rainwater it would add a hydrostatic thrust $\tfrac12\gamma_w z_0^2 = 44.1$ kN/m acting at $z_0/3$ above D, which would reduce the factor of safety by a further 4–5 %.
Canal filled to the top of the bank. When the water level outside the slope is the same as the water level inside it, the water exerts no net seepage force and its only effect is buoyancy: every element of the sliding mass now weighs its submerged weight. The restoring moment is unchanged, because $c_u$ is a total-stress strength that is not altered by uniform submergence, so$$W^{\prime} = A(\gamma - \gamma_w) = 155(20 - 9.81) = 1579.5\ \text{kN/m},$$$$\begin{aligned}M_D^{\prime} &= 1579.5(3.2) = 5054\ \text{kN}\cdot\text{m/m} \\ FS &= \frac{12\,404}{5054}\end{aligned}$$$$\boxed{FS = 2.45 \quad \text{with the canal full}}$$
Interpretation. Filling the canal almost doubles the factor of safety, because the water outside the bank supports it. The engineering consequence is the opposite of what the number suggests: the canal is at its most dangerous not when it is full but when it is emptied quickly. Under rapid drawdown the external water is removed while the clay is still saturated and still carrying the pore pressures of the full condition, so the disturbing moment returns to the full-weight value while the effective strength has not yet recovered — the critical case for design, and the reason canal and reservoir banks are drawn down slowly or provided with drainage blankets. A tension crack filled with water during drawdown makes the same case worse.
