(a) A vertical cut, 4·5 m deep, is to be made in a c-φ soil having cohesion = 19·1 kN/m², angle of internal friction = 16° and unit weight = 18·5 kN/m³. Compute the following :

(i) The active earth pressure at the top and bottom of the cut

(ii) The

Question

(a) A vertical cut, 4·5 m deep, is to be made in a c-φ soil having cohesion = 19·1 kN/m², angle of internal friction = 16° and unit weight = 18·5 kN/m³. Compute the following :

(i) The active earth pressure at the top and bottom of the cut

(ii) The depth up to which the tension cracks develop

(iii) The maximum depth of excavation that can be left unsupported

15

(b) A vertical sluice gate with an opening of 0·60 m produces a downstream jet with a depth of 0·40 m when installed in a long rectangular channel, 5·0 m wide, conveying a steady discharge of 20 m³/s. It is observed that the flow, downstream of the gate eventually returns to a uniform depth of 2·5 m. Indicate whether jump will occur or not. Justify the answer. Calculate the following :

(i) Energy head loss

(ii) Upstream depth

(iii) Force on the gate

Briefly explain the applications of hydraulic jump. Can we apply critical energy concept in case of hydraulic jump? Justify your answer. 20

(c) The following data was obtained from a plate load test carried out on a 60 cm square test plate at a depth of 2 m below the ground surface on a sandy soil with water table at a great depth :

| Load intensity (kN/m²) | 0 | 50 | 100 | 150 | 200 | 250 | 300 |
|------------------------|---|----|-----|-----|-----|-----|-----|
| Settlement (mm)        | 0 | 2·0| 4·0 | 7·5 | 11·0| 16·3| 23·5|

Determine the settlement of a 3 m × 3 m square footing founded at a depth of 2 m below the ground surface, carrying a load of 1100 kN, and compare this settlement with the permissible settlement specified by the Indian Standards. 15

UPSC Answer Check · Accepted Answer

Begin with a brief introduction stating the governing theories (Rankine's earth pressure, momentum principles for hydraulic jump, and Terzaghi's consolidation for settlement). Allocate approximately 30% time to part (a) on active earth pressure and tension cracks, 40% to part (b) on hydraulic jump analysis and sluice gate forces, and 30% to part (c) on plate load test interpretation using IS 1888 provisions. Present all derivations systematically with clear sub-headings for each sub-part, and conclude with practical implications for Indian field conditions.
- Part (a): Correct application of Rankine's active earth pressure theory for c-φ soil with tension crack depth z₀ = 2c/(γ√Ka) and critical unsupported depth Hc = 4c/(γ√Ka)
- Part (a): Calculation of Ka = tan²(45°-φ/2) = tan²(37°) = 0.568, active pressure at top = -2c√Ka (negative indicates tension), at bottom = γHKa - 2c√Ka
- Part (b): Application of momentum equation and specific energy principles to determine conjugate depths, verification of jump occurrence using Froude number (Fr₁ > 1 and sequent depth y₂ > y₁)
- Part (b): Calculation of energy loss ΔE = (y₂-y₁)³/(4y₁y₂), upstream depth from energy equation, and gate force using momentum flux difference with hydrostatic pressure terms
- Part (b): Explanation of hydraulic jump applications (energy dissipation at dam toe, mixing in water treatment, aeration) and critical analysis of why critical energy concept cannot be applied (energy loss violates constant specific energy assumption)
- Part (c): Application of IS 1888:1982 for plate load test interpretation, use of pressure-settlement curve to determine modulus of subgrade reaction, and extrapolation to prototype footing using influence factor method or Schmertmann's correction for sand

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies Rankine's theory for part (a), momentum principles and conjugate depth relationships for part (b), and IS 1888 methodology with appropriate correction factors for part (c); distinguishes between total and effective stress conditions where relevant	Identifies basic theories but confuses active with passive earth pressure, or misapplies jump conditions, or uses linear extrapolation without correction factors for settlement	Applies wrong theories (Coulomb instead of Rankine, or energy equation instead of momentum for jump), or ignores influence of water table, or uses direct scaling without any correction
Numerical accuracy	25%	12.5	All calculations precise to 2 decimal places: Ka = 0.568, z₀ = 2.87 m, Hc = 5.74 m; jump verified with Fr₁ = 7.07, y₂ = 3.87 m > 2.5 m so jump occurs; energy loss = 2.42 m; settlement ≈ 25-30 mm using proper influence factor	Correct methodology but arithmetic errors in final values, or unit conversion mistakes (kN/m³ vs kN/m²), or incorrect interpolation from load-settlement data	Order of magnitude errors, ignores square root in tension crack formula, or calculates jump sequent depth incorrectly leading to wrong conclusion about jump occurrence
Diagram quality	15%	7.5	Clear pressure distribution diagrams for part (a) showing tension zone and cracked zone; hydraulic jump profile with energy grade line and momentum function diagram for part (b); schematic of plate load test setup with load-settlement curve plotted for part (c)	Basic sketches without proper labeling of axes, missing critical depths, or free-hand curves without scale indication	No diagrams despite visual nature of problems, or completely wrong diagrams (passive pressure instead of active, or gradually varied flow profile instead of jump)
Step-by-step derivation	25%	12.5	Systematic derivation: explicit statement of formulas, substitution of values with units, intermediate results shown, and final answers boxed; for part (b) shows complete momentum balance with all terms; for part (c) shows pressure-settlement relationship and modulus calculation	Some steps skipped or combined, missing unit consistency checks, or jumps from formula to final answer without showing substitution	Only final answers with no working, or incorrect formula manipulation (algebraic errors in expanding terms), or missing critical steps in derivation
Practical interpretation	15%	7.5	Compares calculated settlement with IS 1904 permissible limits (40 mm for isolated footings on sand), discusses need for support in tension crack zone for part (a), explains energy dissipation importance for Indian dam safety in part (b), and comments on scale effects in plate load tests	Generic statements about safety without specific IS code references, or superficial discussion of applications without linking to Indian conditions	No interpretation of results, or irrelevant discussion not connected to calculated values, or dangerous conclusions (suggesting unsupported excavation beyond critical depth)

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