(a) A 300 mm diameter concrete pile is to be driven into a medium dense to dense sand with an embedded length of 12 m. The soil conditions are shown in the figure. No ground water was encountered and the ground water table is not expected to rise dur

Question

(a) A 300 mm diameter concrete pile is to be driven into a medium dense to dense sand with an embedded length of 12 m. The soil conditions are shown in the figure. No ground water was encountered and the ground water table is not expected to rise during the life of the structure. Given: The coefficient lateral earth pressure (k) = 0·95, tan δ = 0·45 and for φ = 38° bearing capacity factor, N_q = 80. Determine the pile's axial capacity with a factor of safety of 2. Assume critical depth of the pile is equal to 20 times the diameter of the pile. (15 marks)

(b) A Pelton wheel develops 5520 kW power under a head of 240 m at an overall efficiency of 80% when revolving at a speed of 200 rpm. Find the unit discharge, unit power and unit speed. Assume peripheral coefficient as 0·46. If the head on the same turbine falls during summer season to 150 m, find the discharge, power and speed at this head. (15 marks)

(c) A 1 m long hollow shaft is to transmit a torque of 400 N-m. The outer diameter of the shaft must be 25 mm to fit existing attachments. The relative rotation of the two ends of the shaft is limited to 0·375 rad. The shaft can be made of either titanium alloy or aluminium. Using the data given in the table below, determine the maximum inner radius to the nearest millimeter of the lightest shaft that can be used for transmitting the torque. (20 marks)

| Material | Shear Modulus G (GPa) | Maximum Shear Stress τₐₗₗₒw (MPa) | γ (density) (Mg/m³) |
|----------|----------------------|-----------------------------------|---------------------|
| Titanium alloy | 36 | 450 | 4·4 |
| Aluminium | 28 | 150 | 2·8 |

UPSC Answer Check · Accepted Answer

Calculate the required quantities for all three sub-parts systematically. For part (a), compute end bearing and skin friction resistance of pile using given soil parameters and apply factor of safety. For part (b), determine unit quantities first, then use similarity laws to find summer conditions. For part (c), apply torsion formulas considering both stress and angle of twist constraints to find optimal inner radius for minimum weight. Allocate approximately 30% time to (a), 30% to (b), and 40% to (c) based on marks distribution.
- Part (a): Calculate ultimate pile capacity using Q_u = Q_b + Q_s with end bearing (q_b = σ'_v × N_q) and skin friction (f_s = k × σ'_v × tan δ) considering critical depth limitation of 20D = 6 m for stress calculation
- Part (a): Apply factor of safety of 2 to obtain safe axial capacity, recognizing that 12 m embedment exceeds critical depth so vertical stress increases only up to 6 m
- Part (b): Calculate unit quantities (Q_u, P_u, N_u) using P = γ_w × Q × H × η_o and similarity laws, then determine discharge, power and speed at H = 150 m using Q ∝ H^0.5, P ∝ H^1.5, N ∝ H^0.5
- Part (c): Establish governing equations for hollow shaft: τ_max = T×r_o/J ≤ τ_allow and θ = TL/(GJ) ≤ 0.375 rad, where J = π(r_o^4 - r_i^4)/2
- Part (c): Solve for minimum inner radius satisfying both constraints for each material, then compare weights W = γ×π(r_o^2 - r_i^2)×L to select lightest shaft
- Part (c): Recognize that for titanium alloy, stress constraint governs (high τ_allow), while for aluminium, angle of twist constraint governs (lower G), requiring iterative check of both conditions

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies critical depth concept for pile capacity (part a), applies similarity laws and unit quantities for turbines (part b), and recognizes dual constraints (stress and twist) with correct polar moment expressions for hollow shafts (part c)	Uses basic formulas correctly but misses critical depth limitation in (a), confuses unit quantity definitions in (b), or applies solid shaft formulas to hollow shaft in (c)	Fundamental errors like using total stress instead of effective stress for pile capacity, incorrect similarity law exponents for turbine scaling, or wrong torsion formula application
Numerical accuracy	25%	12.5	All calculations precise with correct unit conversions (MPa to Pa, GPa to Pa, Mg/m³ to kg/m³), proper handling of critical depth in (a), accurate similarity law computations in (b), and correct iterative solution for inner radius to nearest mm in (c)	Minor arithmetic errors or unit conversion mistakes, approximate values for intermediate steps, or rounding errors in final answers	Major calculation errors, wrong order of magnitude results, missing factor of safety application, or completely incorrect numerical substitutions
Diagram quality	10%	5	Clear sketch of pile-soil system showing critical depth and stress distribution for (a), velocity triangles or working schematic for Pelton wheel in (b), and cross-sectional diagram of hollow shaft with dimensions for (c)	Basic diagrams present but lacking labels or incomplete representation of key features like critical depth marking or shaft dimensions	No diagrams, or diagrams that misrepresent the physical systems (e.g., wrong pile embedment, incorrect turbine type, solid instead of hollow shaft)
Step-by-step derivation	25%	12.5	Logical progression showing all steps: effective stress calculation with depth limitation, unit quantity derivations, explicit J expression expansion, and systematic comparison of both materials with clear statement of governing constraint for each	Steps shown but with gaps in derivation, jumps between equations without explanation, or missing justification for selecting final answer from multiple constraints	Final answers stated without derivation, incorrect formula substitutions, or missing essential steps like unit quantity calculation before similarity law application
Practical interpretation	20%	10	Interprets results practically: comments on pile capacity adequacy for typical loads, discusses turbine performance variation with seasonal head changes relevant to Indian hydro projects (e.g., Himalayan rivers), and justifies material selection based on weight-critical applications like aerospace or automotive	Brief mention of practical implications without specific context, generic statements about safety or efficiency without linking to calculated values	No interpretation of results, or irrelevant discussion not connected to calculated values; fails to recognize that titanium's higher cost may be justified by weight savings despite higher material expense

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