(a) The figure below represents time and consolidation relationship for a clay sample 30 mm thick subjected to a given pressure range under double drainage condition.

Determine :

(i) The coefficient of consolidation (Cᵥ) for the sample.

(ii) The t

Question

(a) The figure below represents time and consolidation relationship for a clay sample 30 mm thick subjected to a given pressure range under double drainage condition.

Determine :

(i) The coefficient of consolidation (Cᵥ) for the sample.

(ii) The time required for 75% consolidation of the same clay soil, if it were 2 m thick with similar drainage condition.

(iii) The time required for same degree of consolidation with single drainage condition.

Given :
T = π/4 U²   U < 60%
T = (-) 0·933 log₁₀ (1 - U) - 0·085   U > 60%

(b) A single angle strut ISA 80 × 80 × 10 is used to carry a service load of 80 kN. The centre to centre distance between the end connections is 2 m. The end connection is done by two bolts. Check the adequacy of the section to carry this load.

The grade of steel is E 250. Use limit state method.
Take K₁ = 0·2, K₂ = 0·35 and K₃ = 20 for 'fixed' fixity as per code IS 800 : 2007.

Properties of ISA 80 × 80 × 10
A = 1500 mm²
rᵧ = 24·1 mm
rᵤ = 24·1 mm
rᵤᵤ = 30·4 mm
rᵥᵥ = 15·5 mm

(c) A conical draft tube having inlet and outlet diameters 1 m and 1·5 m discharges water at outlet with a velocity of 2·5 m/s. The total length of the draft tube is 6 m and 1·2 m of the length of draft tube is immersed in water. If the atmospheric pressure head is 10·3 m of water and loss of head due to friction in the draft tube is equal to 0·20 times the velocity head at outlet of the tube, find :

(i) Pressure head at inlet

(ii) Efficiency of draft tube

UPSC Answer Check · Accepted Answer

Calculate numerical solutions for all six sub-parts systematically. For part (a), extract data from the time-consolidation curve to find Cᵥ using Tv = Cv·t/H², then apply time factor scaling for thickness and drainage changes. For part (b), check slenderness ratio, apply IS 800:2007 buckling provisions with given K-factors, and verify design compressive stress against factored load. For part (c), apply Bernoulli's equation between inlet and outlet, accounting for velocity head conversion, friction loss (0.2×V₂²/2g), and draft tube immersion effects. Allocate approximately 35% effort to part (a) due to curve interpretation complexity, 35% to part (b) for code-based calculations, and 30% to part (c). Present all derivations with proper unit conversions and final answers with appropriate significant figures.
- Part (a)(i): Correctly read t₅₀ or t₉₀ from the time-consolidation curve, calculate Tv using given equations, and solve Cv = Tv·H²/t with H = 15 mm (half-thickness for double drainage)
- Part (a)(ii): Apply time factor proportionality t ∝ H² with same U=75%, using new H = 1000 mm (half-thickness for 2m double drainage) to find required time
- Part (a)(iii): Recognize single drainage doubles drainage path (H = 2000 mm), hence time increases by factor of 4 compared to double drainage for same consolidation
- Part (b): Calculate effective length KL = 0.85×2000 = 1700 mm (fixed-fixed), slenderness ratios about u-u, v-v axes, interpolate design compressive stress from IS 800 Table 9(c), and check against 1.5×80 = 120 kN factored load
- Part (c)(i): Apply energy equation between inlet (section 1) and outlet (section 2), calculate V₁ by continuity, include friction loss 0.2×V₂²/2g, and solve for p₁/γg accounting for 1.2m submergence
- Part (c)(ii): Calculate efficiency η = (actual kinetic energy recovery)/(ideal kinetic energy recovery) = (V₂²/2g - hf - exit loss)/(V₂²/2g) or equivalent pressure recovery ratio

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies double drainage H = H_total/2 for parts (a)(i)-(ii) but H = H_total for single drainage in (a)(iii); applies IS 800:2007 clause 7.5.1 for angle strut buckling about minor axis; recognizes draft tube functions as kinetic energy recovery device with proper sign convention for pressure head	Uses correct general formulas but makes errors in drainage path identification (e.g., uses full thickness for double drainage) or applies wrong buckling curve; understands Bernoulli's principle but omits submergence effect or friction loss term	Fundamental misconceptions: treats single and double drainage identically, uses Euler buckling formula instead of IS code provisions, or applies hydrostatic pressure equation without energy conservation
Numerical accuracy	25%	12.5	All six numerical answers accurate within 2% tolerance; correct unit conversions (mm to m, kN to N); proper interpolation in IS 800 tables; maintains 3-4 significant figures throughout; correct final answers: Cv ≈ 2-4 m²/year, t₇₅ ≈ 2-5 years for 2m, t_single ≈ 4×t_double, strut capacity > 120 kN, p₁/γg ≈ 2-4 m absolute, η ≈ 60-80%	Correct methodology with calculation errors (10-20% deviation); wrong significant figures; arithmetic mistakes in unit conversion; incorrect interpolation but correct table selection; partial credit for correct setup	Order of magnitude errors; wrong formulas leading to nonsensical answers (negative time, pressure head exceeding atmospheric by factor >10); no unit consistency; missing final numerical answers
Diagram quality	10%	5	Reproduces key features of time-consolidation curve showing U vs √t or t with marked t₅₀/t₉₀ points; sketches double vs single drainage boundary conditions; draws angle strut with end connection details; illustrates draft tube with inlet/outlet sections, datum, and head components	Basic sketches without proper labeling; missing key features like drainage faces or flow direction; incomplete draft tube geometry; adequate but not exemplary	No diagrams despite curve interpretation requirement; confusing or wrong sketches (e.g., single drainage shown as double); missing critical labels for sections 1 and 2
Step-by-step derivation	25%	12.5	Explicit statement of all governing equations (Tv = Cv·t/H², IS 800 design stress formula, Bernoulli's equation with all terms); clear substitution sequence; shows intermediate values (Tv, H, V₁ calculation); justifies K-factor selection (0.85 for fixed-fixed per IS 800); derives continuity equation for draft tube	Correct equations stated but skips key substitution steps; jumps to final answer; shows some work but lacks clarity on which H value used; partial derivation for some parts only	No derivations shown—only final answers; equations written without variables defined; circular reasoning; incorrect equation selection without justification
Practical interpretation	20%	10	Interprets Cv magnitude for typical clay (10⁻⁷ to 10⁻⁸ m²/s range); comments on impracticality of years-long consolidation for 2m layer and need for vertical drains; discusses strut buckling mode preference and connection rigidity; explains draft tube cavitation risk if pressure drops below vapor pressure and efficiency implications for turbine performance	Brief mention of practical significance without elaboration; standard concluding remarks; recognizes but does not develop engineering implications	No interpretation of results; fails to recognize unrealistic answers (e.g., negative pressure heads without cavitation comment); no connection to field applications or design consequences

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