UPSC Civil Engineering 2024 — Paper I

Solve all five sub-parts systematically, allocating approximately 20% time to each part since marks are equally distributed. Begin with clear free body diagrams for part (a), then proceed through numerical calculations for parts (b), (c) and (d) showing all steps, and conclude with a structured explanation for part (e). Present solutions in sequence (a) through (e) with clear headings and proper units throughout.

• Part (a): Correct free body diagrams showing reactions at fixed end A, equilibrium at free end D with applied moment, and internal forces/shear-moment distribution for member BC under UDL
• Part (b): Calculation of direct compressive stress (P/A) and bending stresses (My/I) due to eccentricity at point P, superposition of stresses at corners A, B, C, D using appropriate section properties
• Part (c): Application of influence line concept or moving load analysis to determine maximum BM at 6m from left support, correct positioning of UDL for maximum effect using influence line ordinates
• Part (d): Analysis of bolt group eccentricity, calculation of direct shear and torsional shear, vectorial combination to find resultant force on critical bolt A, comparison with given bolt value
• Part (e): Functions of transverse reinforcement including confinement of core concrete, prevention of longitudinal bar buckling, shear resistance, and ductility enhancement as per IS 456 provisions

Begin with a clear statement of given data for each sub-part. For part (a), apply bending stress formula σ = My/I and shear stress formula τ = VQ/(Ib) systematically. For part (b), derive the neutral axis depth using strain compatibility and equilibrium of forces from first principles with proper stress-strain diagrams. For part (c), design weld lengths using IS 800:2007 provisions for eccentrically loaded welded connections. Allocate approximately 40% time to part (a), 25% to part (b), and 35% to part (c) based on marks distribution.

• Part (a): Correct identification of maximum bending moment location and value for cantilever under UDL; calculation of section modulus; determination of safe UDL using σ = M/Z; computation of maximum shear stress at support using τ = VQ/(Ib) or simplified formula
• Part (a): Proper handling of beam geometry (ABC with possible overhang or varying section) and correct interpretation of 'maximum safe allowable bending stress' as working stress
• Part (b): Derivation starting from strain distribution (linear, εc at top, εs at steel level), stress-strain relationship for concrete (parabolic-rectangular or as per given Fig. 1), stress-strain for steel (bilinear or as per given Fig. 2)
• Part (b): Force equilibrium equation: Cc = T → 0.36fckbxu = 0.87fyAst; solving for neutral axis depth xu = (0.87fyAst)/(0.36fckb); clear definition of all terms
• Part (c): Identification of weld configuration (two angles, welds on both sides of gusset); calculation of throat thickness = 0.7×size; determination of weld strength per unit length = fu/(√3×γmw); resolution of force components and moments about C.G.
• Part (c): Application of IS 800:2007 clause on eccentrically loaded welds: calculation of direct shear and torsional shear, vectorial combination, and iterative or direct solution for lengths l₁ and l₂ satisfying strength requirements

Calculate the required quantities across all three sub-parts, allocating approximately 40% of effort to part (a) Moment Distribution Method (20 marks), 40% to part (b) T-beam flexural design (20 marks), and 20% to part (c) pipe deflection (10 marks). Begin with clear identification of given data, proceed through systematic calculations with proper formulae from IS 456:2000 and structural mechanics, and conclude with final answers and sketches where demanded. For part (a), show the complete moment distribution table; for part (b), check neutral axis position and design accordingly; for part (c), apply standard deflection formula for UDL.

• Part (a): Correct application of Moment Distribution Method with proper distribution factors, carry-over factors, and iterative balancing to obtain final end moments for box culvert frame ABCD with given FEM values
• Part (a): Accurate sketch of bending moment diagram showing hogging and sagging moments at corners and mid-spans of the box culvert
• Part (b): Correct determination of neutral axis depth by comparing xu with Df using IS 456:2000 Clause 23.1.1, identifying whether flange is in tension or compression zone
• Part (b): Proper calculation of limiting moment of resistance and required steel area using appropriate equations for T-beam (xu < Df or xu > Df case), with final provision of reinforcement bars
• Part (c): Correct calculation of moment of inertia for hollow circular section (pipe) using I = π/64 × (D⁴ - d⁴), and application of δmax = 5wL⁴/(384EI) for simply supported UDL
• Part (c): Proper unit conversion (GPa to kN/m², mm to m) and final deflection value in mm with appropriate significant figures

Solve this multi-part numerical and descriptive question by allocating approximately 40% time to part (a) prestressed concrete calculations, 35% to part (b) truss deflection analysis, and 25% to part (c) steel tension member failure modes. Begin with clear sectional property calculations for the T-beam, apply Pigeaud's or relevant prestress theory with stress constraints, then use unit load method systematically for the truss, and conclude with well-labelled sketches for failure modes. Present each part distinctly with proper headings and maintain sequential logical flow from given data to final results.

• Part (a): Calculate section properties of T-beam (area, centroid, moment of inertia) considering flange 1500×200 mm and rib 300×1200 mm; determine dead load as 25 kN/m³ × cross-sectional area
• Part (a): Apply stress equations at top and bottom fibres using Pᵢ and e with 16% loss factor, setting σ_top = 0 and σ_bottom = 5 MPa under working loads to solve simultaneous equations
• Part (b): Identify zero-force members and calculate member forces under actual loading and unit load at joint C using method of joints or sections
• Part (b): Apply unit load method formula Δ = Σ(NnL)/(AE) for all members, tabulating forces N (actual), n (unit), lengths L, and summing contributions
• Part (c): Enumerate four failure modes: gross section yielding, net section rupture, block shear failure, and shear lag effects with end connections
• Part (c): Draw clear sketches showing each failure mode with failure planes marked, particularly for bolted/riveted connections typical in Indian bridge girders

Calculate requires systematic numerical problem-solving across all five sub-parts. Allocate approximately 20% time to each sub-part (a-e) as they carry equal marks. Begin each sub-part by stating the governing formula, substitute values with proper unit conversions, show intermediate calculations, and conclude with final answers in correct SI units. For sub-part (c), note that the figure is not provided—state reasonable assumptions about the pipe configuration (likely vertical with tank at top). No separate introduction or conclusion is needed; present each sub-part clearly labelled with complete working.

• Sub-part (a): Apply viscous torque formula T = μ(2πR³ωL)/h for concentric cylinder viscometer; convert rpm to rad/s and solve for dynamic viscosity μ
• Sub-part (b): Determine Reynolds number to confirm laminar flow; apply Blasius solution or appropriate flat plate boundary layer equations for drag, boundary layer thickness δ, and wall shear stress τw
• Sub-part (c): Apply Hagen-Poiseuille equation for laminar pipe flow considering hydrostatic head as driving pressure; verify laminar assumption with Reynolds number
• Sub-part (d): Calculate bulk density from core-cutter data, moisture content from oven-drying, then derive dry density, void ratio using Gs, and degree of saturation
• Sub-part (e): Apply plate load test scaling laws for sandy soils (settlement proportional to plate width); extrapolate from 300mm and 600mm plates to 1.5m footing using Terzaghi's settlement relationship

Solve all three sub-parts systematically, allocating approximately 35% time to part (a) dimensional analysis, 30% to part (b) potential flow calculations, and 35% to part (c) pile group capacity. Begin each part with stating the governing equations, show complete derivations with proper units, and conclude with boxed final answers. For part (c), explicitly state whether block failure or individual pile failure governs.

• Part (a): Apply Buckingham π-theorem correctly with 6 variables and 3 fundamental dimensions to obtain two independent dimensionless groups (Froude number and Reynolds number based forms)
• Part (b): Calculate velocity components u = ∂ψ/∂y = 2x and v = -∂ψ/∂x = -2y, then find velocity magnitude and direction at P(2,3); verify irrotationality and obtain φ = x² - y² + C
• Part (c): Calculate individual pile capacity using Qs = α·c·As with appropriate adhesion factor; determine group efficiency using Converse-Labarre formula or block failure perimeter; compare block failure vs. individual failure modes
• Part (c): Correctly apply spacing parameters (s = 1.0 m, s' = 0.75 m) and group dimensions (2.5 m × 2.0 m block) for efficiency and block failure calculations
• Part (c): Apply factor of safety of 2.5 to the lesser of group capacity or sum of individual pile capacities divided by efficiency

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

Solve this multi-part numerical problem by allocating approximately 30% time to part (a) passive earth pressure, 30% to part (b) Mohr-Coulomb failure analysis, and 40% to part (c) turbine calculations. Begin each part with the relevant governing equation, show complete derivations with proper units, draw required diagrams to scale with clear labeling, and conclude with physically verified numerical answers.

• Part (a): Calculate passive earth pressure coefficient Kp using Rankine theory, determine pressure distribution with surcharge component, draw trapezoidal pressure diagram, and locate centroid for resultant thrust application point
• Part (b): Establish Mohr-Coulomb failure envelope from given stress state, determine cohesion c and friction angle φ, then calculate major principal stress for modified minor principal stress conditions for both φ = 35° and φ = 0° cases
• Part (c)(i): Calculate power developed using P = η₀ × ρgQH with proper unit conversion to kW
• Part (c)(ii): Determine inlet diameter from peripheral velocity u₁ = πD₁N/60 and width from continuity equation Q = πD₁B₁Vf₁
• Part (c)(iii): Find guide blade angle α from tanα = Vf₁/u₁ using velocity triangle geometry
• Part (c)(iv): Calculate runner inlet angle β from tanβ = Vf₁/(u₁ - Vw₁) using hydraulic efficiency to find Vw₁
• Verify all calculations: passive pressure increases with depth, failure envelope is consistent, turbine velocities satisfy √2gH relationships, and radial discharge implies Vw₂ = 0