18-Env-A3 Geotechnical and Hydrogeological Engineering · December 2019
Question 6 of 6: Factor of Safety of a Cut Slope (Culmann's Method)
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams December 2019 — 18-Env-A3, Geotechnical and Hydrogeological Engineering — 3 hours, open book. All SIX questions are answered below (the exam marks only the first five as submitted; all six are solved here as a complete study resource). Marking scheme: each question 20 marks, equal value.
Given. A homogeneous $c$–$\varphi$ soil slope cut through the toe.
Given data
Quantity
Value
Cut height, $H$
10 m
Unit weight, $\gamma$
18 kN/m³
Angle of internal friction, $\varphi$
20°
Cohesion, $c$
30 kN/m²
Find. (a) FS at $\beta=35^\circ$. (b) Maximum $\beta$ for $\text{FS}\ge1.5$. (c) Additional measures for FS > 3 at $\beta=35^\circ$.
Figure 4: cut slope of height $H$ at angle $\beta$ through the toe, in a $c$–$\varphi$ soil.
Approach. Use Culmann's plane-failure method with the factor of safety applied simultaneously to both strength components ($c_d=c/\text{FS}$, $\tan\varphi_d=\tan\varphi/\text{FS}$), so that the slope's critical height under the mobilized (reduced) strength exactly equals the actual 10 m cut; solve implicitly for FS at a given $\beta$, and for $\beta$ at a given target FS.
Governing equation. Culmann's critical-height formula, evaluated with mobilized strength $c_d=c/\text{FS}$ and $\varphi_d=\arctan(\tan\varphi/\text{FS})$, must equal the actual height $H$: $$H = \dfrac{4c_d\sin\beta\cos\varphi_d}{\gamma\left[1-\cos(\beta-\varphi_d)\right]}$$ This is implicit in FS (through $\varphi_d$), so it is solved numerically (bisection) for each part below.
(a) Factor of safety at $\beta=35^\circ$. Solving the governing equation for FS with $\beta=35^\circ$ fixed gives $$\boxed{\text{FS} = 3.15}$$ (cross-check: the simpler, unreduced-strength ratio $H_{cr}(\varphi,c)/H=105.5/10=10.5$ is not used here, since it applies full, un-factored shear strength to the trial surface and so overstates the safety margin – the reduced-strength form above is the standard definition of FS for a $c$–$\varphi$ slope.)
(b) Maximum $\beta$ for $\text{FS}=1.5$. Solving the same governing equation for $\beta$ with $\text{FS}=1.5$ fixed: $$\boxed{\beta_{max} \approx 66.5^\circ}$$ FS decreases steadily as the cut is steepened (from 3.15 at $35^\circ$ down toward 1.0 near vertical), so any $\beta$ up to about $66.5^\circ$ keeps $\text{FS}\ge1.5$.
(c) Additional measures for $\text{FS}>3$ at $\beta=35^\circ$. FS is already 3.15 at $35^\circ$ – only marginally above 3 – so a design would want real margin, not a bare pass. Practical measures: install toe and face subdrains (or horizontal drains) to keep pore pressures low and prevent strength loss from softening/seepage; bench the cut into two or three lifts with berms rather than one continuous face, which reduces the effective driving wedge; add reinforcement (soil nails, geogrid, or a retaining/reinforced-earth structure) at the toe; keep surcharge (stockpiles, equipment, structures) away from the crest; and armour/vegetate the face to control surface erosion, which over time can undercut the toe and reduce FS below the design value.
