(a) A canal is to be excavated through a soil with c = 20 kN/m², φ = 20°, e = 0·80 and G = 2·70. The side slope is 1 in 1. The depth of the canal is to be 8 m. Determine the factor of safety with respect to cohesion when the canal runs full. What wil

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(a) A canal is to be excavated through a soil with c = 20 kN/m², φ = 20°, e = 0·80 and G = 2·70. The side slope is 1 in 1. The depth of the canal is to be 8 m. Determine the factor of safety with respect to cohesion when the canal runs full. What will be the factor of safety if the canal is rapidly emptied ? For β = 45°, the stability number for various φ values are as given above. (15 marks)

(b) The soil profile at a building site consists of dense sand up to 3 m depth, normally loaded soft clay from 3 m to 8 m depth and stiff impervious rock below 8 m depth. The ground water table is at 0·60 m depth below ground level. The sand has a density of 18·6 kN/m³ above water table and 19·2 kN/m³ below water table. For the clay, natural water content is 50%, liquid limit is 70% and specific gravity is 2·70. Calculate the probable ultimate settlement resulting from a uniformly distributed surface load of 50 kN/m² applied over an extensive area of the site. (15 marks)

(c) The flow of water in a canal varies from 425 l/s to 680 l/s. It is desired to discharge not less than 340 l/s of water and not more than 425 l/s over a 90° V-notch weir into one channel, while the remainder goes over a sharp-crested rectangular weir. Find the length of rectangular weir and maximum head on each weir. Take C_d = 0·58 for both weirs. (20 marks)

UPSC Answer Check · Accepted Answer

Solve this three-part numerical problem by addressing each sub-question systematically: (a) apply Taylor's stability number method for canal slope stability under full and rapid drawdown conditions, (b) calculate consolidation settlement using compression index correlations for NC clay, and (c) solve simultaneous weir equations for proportional flow division. Present clear sectional diagrams, state all assumptions, and conclude with practical implications for canal design and foundation safety.
- Part (a): Correct use of Taylor's stability number Sn = c/(γH·Fc) with submerged unit weight for full canal and saturated unit weight for rapid drawdown condition
- Part (a): Calculation of factor of safety with respect to cohesion for both cases using given β=45° and φ=20°
- Part (b): Determination of compression index Cc from liquid limit (Cc = 0.009(LL-10)), initial void ratio from water content, and effective stress increase in clay layer
- Part (b): Application of consolidation settlement formula ΔH = (Cc·H/(1+e₀))·log₁₀((σ'₀+Δσ')/σ'₀) for normally consolidated clay
- Part (c): Setting up discharge equations: Qv = Cd·(8/15)·√(2g)·tan(θ/2)·H^(5/2) for V-notch and Qr = Cd·(2/3)·√(2g)·L·H^(3/2) for rectangular weir with simultaneous solution

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	25%	12.5	Correctly identifies Taylor's stability number concept, distinguishes between submerged vs. saturated unit weights for drawdown analysis, applies Skempton's Cc correlation appropriately, and sets up proper weir discharge relationships with correct exponents	Uses stability number method but confuses unit weights or makes minor errors in Cc correlation; weir equations mostly correct but may miss tan(45°)=1 simplification	Misapplies stability number formula, uses wrong unit weights, employs incorrect settlement formula (e.g., uses e instead of 1+e), or fundamentally wrong weir discharge equations
Numerical accuracy	25%	12.5	All calculations precise with correct unit conversions (kN/m³, m/s²), accurate logarithmic evaluations, and final answers within 2% of expected values: Fc≈1.4-1.6 (full), Fc≈0.9-1.1 (drawdown), settlement≈350-450mm, rectangular weir length≈0.8-1.0m, heads≈0.35-0.55m	Correct methodology but arithmetic errors leading to 5-10% deviation in final answers; minor unit conversion errors	Major calculation errors, wrong order of magnitude, or missing critical steps leading to unrealistic results
Diagram quality	15%	7.5	Clear labeled diagrams showing: (a) canal cross-section with slip circle, tension crack, and forces; (b) soil profile with stress distribution diagram showing effective stress increase with depth; (c) weir arrangement with flow distribution and head definitions	Basic sketches present but missing key labels or dimensions; free-body diagrams incomplete	No diagrams or completely unrecognizable sketches; missing critical elements like slip surface or stress diagrams
Step-by-step derivation	20%	10	Systematic presentation: states given data, shows formula with substitution, calculates intermediate steps (γsat, γsub, σ'₀ at mid-clay), and clearly separates the three parts with proper headings and logical flow	Some steps shown but jumps between calculations; missing intermediate values or unclear logical progression	Disorganized working with no clear steps; direct final answers without showing methodology; mixed-up parts
Practical interpretation	15%	7.5	Interprets Fc<1 for rapid drawdown as critical failure condition requiring staged dewatering; discusses settlement magnitude relevance for building foundations on NC clay (e.g., Bengal basin deposits); explains weir design ensures minimum ecological flow in diversion canals	Brief mention of implications without depth; generic statements about safety without specific context	No interpretation or completely irrelevant comments; fails to recognize unsafe conditions or practical design significance

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