UPSC Civil Engineering 2021

This is a multi-part numerical problem requiring systematic solution of five independent engineering calculations. Allocate approximately 20% time to each sub-part (a-e) as marks are equal. Begin each part with clear identification of the governing principle, show all formulae with standard notations, substitute values with units, and present final answers with proper sign conventions and units. No introduction or conclusion is needed; use clear sectional headings for each part.

• Part (a): Correct application of method of joints or method of sections to determine forces in members AC, DE and GH; identification of zero-force members if applicable; proper sign convention for tension/compression
• Part (b): Calculation of prestressing force eccentricity; determination of moment due to external UDL; application of P/A ± Pe/Z ± M/Z formula; correct stress values at top and bottom fibres
• Part (c): Design of fillet weld connection considering direct shear and torsional shear; calculation of throat thickness; weld length determination; check against permissible stress as per IS 800
• Part (d): Application of Newton's law of viscosity for parallel plate flow; calculation of shear stress and force for symmetric and asymmetric plate positions; correct gap dimensions for each case
• Part (e): Determination of field density, moisture content, and dry density; calculation of minimum and maximum dry densities from loose and compacted states; application of relative density formula as per IS 2720

This question demands solving three distinct numerical problems covering structural analysis, R.C.C. design, and open channel hydraulics. Begin with clear identification of given data for each part, present systematic calculations with appropriate formulae, and conclude with final answers and reinforcement detailing for part (b).

• Part (a): Correct application of moment area theorems with proper handling of internal hinge at B, including M/EI diagram construction and tangential deviation calculations
• Part (b): Calculation of effective length factor (0.8L for given end conditions), slenderness ratio check, minimum eccentricity verification, and longitudinal reinforcement design using SP-16 or working formula with proper lateral ties
• Part (c): Application of Chezy's formula V = C√(RS) for slope determination, conveyance K = AC√R calculation, and Froude number computation to classify flow regime
• Proper use of IS 456:2000 provisions for column design including minimum reinforcement percentage (0.8%) and maximum (4%) checks
• Correct unit conversions throughout (litres/sec to m³/s, mm to m) and consistent use of SI units in final answers
• Neat reinforcement detailing showing longitudinal bars, lateral ties with spacing as per IS 456, and clear cover specifications

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

Calculate the required parameters for all three sub-parts systematically. For (a), apply influence line method or method of sections with load positioning to find maximum tensile force in DI; spend ~35% time here as it involves moving load analysis. For (b), use parallel plate flow equations for laminar flow; allocate ~25% time. For (c), perform complete stability analysis with Rankine's theory; allocate ~40% time as it carries the highest conceptual and calculation load. Present each part with clear headings, free-body diagrams, and final safety checks.

• For (a): Correct influence line construction for member DI or proper method of sections application with critical load positioning (loads moving G to A); identification of maximum tensile force location using Muller-Breslau principle or direct calculation
• For (a): Proper handling of three moving loads (10 kN, 20 kN, 15 kN) with correct load placement for maximum effect on tension member DI
• For (b): Application of laminar flow between parallel plates (Couette/Poiseuille flow) with parabolic velocity profile; correct use of μ = 2.453 N-s/m², h = 0.10 m, u_max = 1.5 m/s
• For (b): Calculation of discharge per unit width (q), wall shear stress (τ_w), pressure gradient (Δp/L), velocity gradients (du/dy) at plates, and velocity at y = 0.02 m using u = u_max[1 - (y/(h/2))²]
• For (c): Rankine's active earth pressure coefficient K_a = (1-sinφ)/(1+sinφ) = 0.271 for φ = 35°; calculation of P_a = ½K_aγH² with proper point of application at H/3 from base
• For (c): Complete stability checks—overturning (factor of safety ≥ 2), sliding (FOS ≥ 1.5 with μ = 0.5), and base pressure (eccentricity check, p_max ≤ 150 kPa) with weight of concrete wall components
• For (c): Proper identification of overturning moment (about toe) and restoring moment including self-weight of stem and base slab; correct calculation of resultant location and eccentricity e = B/2 - x̄

Solve all five sub-parts systematically, allocating approximately 20% time to each part since all carry equal marks. Begin with free-body diagrams for (a), (c), and (e); apply limit state design principles for (b); use dimensional analysis for (d). Present solutions with clear headings, numbered steps, and boxed final answers for each sub-part.

• For (a): Apply Lami's theorem or moment equilibrium about P to find angle 2θ for minimum tension; derive condition dT/dθ = 0 and solve for θ ≈ 45° giving 2θ = 90°, with T_min ≈ 707 N
• For (b): Calculate effective flange width as per IS 456, determine neutral axis depth using strain compatibility, check if section is under-reinforced, and compute moment of resistance using M_u = 0.36 f_ck b x_u (d - 0.42 x_u)
• For (c)(i) and (ii): Apply compatibility of deformations (total elongation = 0) and equilibrium to find reactions R_A and R_D; identify middle segment BC as compression member with force = R_A - P_B or R_D - P_C
• For (d): Apply Froude scaling laws for geometric similarity; scale ratio L_r = 4, so Q_model = Q_prototype × L_r^(5/2) = 3.2 m³/s and ΔP_model = ΔP_prototype × L_r = 400 kN/m² (or using Reynolds similarity if applicable)
• For (e): Apply Mohr-Coulomb failure criterion for cohesionless soil: σ₁ = σ₃ tan²(45°+φ/2); calculate σ₁ ≈ 315 kPa and deviator stress Δσ = σ₁ - σ₃ ≈ 210 kPa

Solve all three parts systematically, allocating approximately 25% time to part (a) on boundary layer power calculation, 40% to part (b) on SFD and BMD analysis with complete diagrams, and 35% to part (c) on limit state design of the RCC beam. Begin with clear identification of given data for each part, show all formulae with IS code references where applicable, present step-by-step calculations, and conclude with final answers in proper units.

• Part (a): Calculate Reynolds number to determine flow regime, select appropriate drag coefficient for cylindrical body, compute wetted surface area, and apply boundary layer drag formula to find power required
• Part (b)(i): Compute support reactions using equilibrium equations (ΣV=0, ΣM=0) considering UDL, point loads and applied moment
• Part (b)(ii)-(iv): Calculate SF and BM values at all salient points (A-F), complete the table, and draw SFD/BMD with proper sign conventions indicating linear/parabolic nature of curves
• Part (c): Check for limiting moment of resistance, calculate design moment, determine required steel area using limit state equations, check for minimum and maximum reinforcement, and provide detailing
• Apply correct IS 456:2000 provisions for limit state design in part (c) including partial safety factors and stress block parameters
• Use proper units throughout (kN, kNm, mm, N/mm²) and maintain consistency in calculations

Solve this three-part numerical problem by allocating approximately 30% time to part (a) on shear strength calculation, 30% to part (b) on bearing capacity with interpolation of bearing capacity factors, and 40% to part (c) on steel beam design including adequacy check and redesign with cover plates. Begin each part with stated assumptions and formulas, show complete step-by-step calculations with proper units, and conclude with clear final answers and practical implications for field application.

• Part (a): Calculate effective stress at mid-depth of sand layer (6m below ground), determine saturated unit weight using void ratio from dry unit weight, compute effective stress considering water table at 2.5m, and apply τ = σ' tan φ for shear strength
• Part (b): Apply Terzaghi's bearing capacity equation for square footing with interpolation of Nc, Nq, Nγ values for φ=21°, use submerged unit weight below water table, calculate ultimate bearing capacity, and determine allowable load with FOS=3
• Part (c): Calculate maximum factored moment for simply supported beam with central point load, check section adequacy using plastic moment capacity (Md = βb·Zp·fy/γm0), and if unsafe, design cover plates to increase section modulus to required value
• Correct application of effective stress principle in parts (a) and (b) with proper handling of water table effects on unit weights and stress calculations
• Proper interpolation technique for bearing capacity factors between φ=20° and φ=22° in part (b), and correct application of IS 800:2007 limit state provisions for plastic section in part (c)
• Appropriate selection and sizing of cover plates in redesign, checking for local buckling criteria and ensuring adequate weld/connection provisions

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

This multi-part question requires solving five distinct technical problems across building construction, project management, surveying, highway engineering, and railway engineering. Allocate approximately 2 minutes per mark (20 minutes per part), presenting each sub-part (a) through (e) as separate, clearly labelled sections. Begin with direct answers for theoretical parts, show complete numerical working for calculation-based parts, and ensure diagrams are neatly drawn with proper labelling.

• Part (a): Purpose of first coat plastering (keying surface, leveling, damp-proofing); formwork loads (fresh concrete weight, construction live loads, wind, vibration, impact); correct identification of pitched roof components (rafter, purlin, tie beam, king post/queen post)
• Part (b): Correct conversion from A-O-A to A-O-N network topology; accurate forward and backward pass calculations; identification of critical path with zero float activities
• Part (c): Application of sensitivity formula n = (S/R) × 206265 seconds; correct calculation of angle change from bubble displacement; derivation of radius of curvature R = S × D / α in arc-seconds
• Part (d): Definition of vertical curve as transition curve in elevation; significance for sight distance, comfort, drainage; correct geometric sketches for both summit curves (convex) and valley curves (concave) with tangent lengths
• Part (e): Application of Indian Railways tractive effort formula considering curve resistance (0.01° per degree), friction limitation, and speed-dependent factors; correct unit conversions for metre gauge conditions

This question demands solving three distinct numerical problems covering resource allocation in project management, highway geometric design, and surveying calculations. The answer should present each part sequentially with clear problem identification, application of correct formulae, systematic calculations, and final verification of results. Part (a) requires resource leveling/ smoothing with Gantt charts or histograms, Part (b) needs set-back distance calculations using IRC standards, and Part (c) involves reciprocal leveling and collimation error determination.

• Part (a): Resource allocation with unlimited resources (resource leveling) and limited resources (resource smoothing) using histograms, identification of peak demand and resource constraints
• Part (a): Clear advantages and disadvantages of both methods including cost implications, project duration impact, and resource utilization efficiency
• Part (b): Correct application of set-back distance formulae for SSD (m < R) and OSD cases, accounting for lane offset of 1.9 m from center line
• Part (c): Calculation of true staff readings using reciprocal leveling principles, elimination of collimation error, and determination of R.L. of B
• Part (c): Recognition that instrument at P gives true difference of level, while Q setup reveals collimation error through inconsistent readings

Begin with a brief introduction acknowledging the multi-disciplinary nature of the question spanning railway engineering, geodetic surveying, and infrastructure financing. Allocate approximately 40% effort to part (a) combining drainage sketches with curve clearance calculations, 30% to part (b) for the intervisibility problem with proper profile diagram, and 30% to part (c) for conceptual comparison of PPP models with a neat BOOT structure diagram. Conclude with a synthesis on integrated infrastructure planning.

• Part (a)(i): Surface water removal via side drains, catch water drains, and proper ballast section; sub-surface water removal via cross drains (drainage layer), pipe drains, and inverted filters with neat sectional sketches
• Part (a)(ii): Calculation of extra clearances on curves using IRC formulae: overthrow (C1), end-throw (C2), lean due to superelevation (C3), and shift; application to inside and outside platforms with 840mm height consideration
• Part (b): Application of line of sight formula considering earth's curvature and refraction (k=0.07); calculation of minimum height of sight line above peaks C and D; determination of non-intervisibility and scaffolding height at B using Indian Survey standards
• Part (c): Definition of infrastructure project with characteristics (public good nature, high capital cost, long gestation, externalities); systematic comparison of BOO vs BOOT on ownership, transfer, financing risk, and government role; typical BOOT structure diagram showing SPV, concessionaire, lenders, and government relationships
• Part (c) continued: Indian examples such as Delhi Metro (PPP variants), NHAI BOT projects, or airport privatization to illustrate BOOT application

Begin with a brief introduction acknowledging the multi-part nature of this pavement design, valuation, and materials question. For part (a), apply IRC:58 design methodology for dowel bars using the given radius of relative stiffness and wheel load, showing complete derivations for bearing, shear, and flexural stress checks. For part (b), prepare a discounted cash flow table year-wise for development costs and revenue realization, computing net present value and cost per m². For part (c)(i), enumerate ferro-cement advantages with specific justification for large tank feasibility, and for (c)(ii) label door components with correct technical terminology. Allocate approximately 40% time to part (a), 30% to part (b), and 30% combined to part (c), ensuring all numerical workings are clearly presented with stated assumptions.

• Part (a): Calculation of load transfer capacity (40% of 5000 kg = 2000 kg), determination of dowel bar diameter using bearing stress criterion (σb = P/(d×h) ≤ 100 kg/cm²), spacing based on relative stiffness (l = 80 cm), and verification against shear (1000 kg/cm²) and flexural (1400 kg/cm²) stress limits with joint width 2.0 cm
• Part (a): Selection of standard dowel bar size (typically 25-32 mm diameter) and spacing (250-300 mm), with 50% load transfer efficiency assumption and check for concrete bearing stress using Bradbury's or IRC:58 recommendations
• Part (b): Year-wise development cost outflow (10%, 15%, 20%, 55% of Rs. 100 × 60,000 = Rs. 6,00,000) and revenue inflow from plot sales (10%, 20%, 20%, 30%, 20% of 60,000 m² at Rs. 5000/m²), discounted at 7% to find NPV and cost per m²
• Part (c)(i): Ferro-cement advantages—thin sections (10-25 mm), high strength-to-weight ratio, crack resistance due to distributed reinforcement, no formwork needed, cost-effectiveness for small tanks; justification for 25,000+ litre tanks requires thickness increase, stiffener rings, and waterproofing treatment
• Part (c)(ii): Correct labeling of door components—styles (vertical), rails (horizontal), mullions (intermediate vertical), panels, stiles, top rail, bottom rail, lock rail, meeting stiles, hinges, latch/lock hardware

Begin with the directive 'calculate' for the numerical parts (a), (b), (c), and (e), while 'discuss' applies to part (d). Allocate approximately 20% time to each sub-part given equal 10-mark weighting. Structure as: direct calculations for (a), (b), (c), (e) with clear steps, followed by a descriptive response for (d) on indicator organisms. No conclusion needed; present each part separately with clear labeling.

• Part (a): Apply superposition principle to 1-hour unit hydrograph; convolute with 70 mm and 40 mm rainfall excesses lagged by 1 hour; identify peak DSR ordinate in m³/s
• Part (b): Apply risk formula R = 1 - (1-1/T)^n where T=25 years, n=3 years; calculate exact probability of overtopping in third year specifically, not cumulative
• Part (c): Use Khosla's theory for exit gradient; apply G_E = (H/d) × (1/π√λ) where λ = (1+√(1+α²))/2 and α = b/d; substitute b=50m, d=5m, H=5m
• Part (d): Define indicator organism; list 4-5 ideal characteristics (e.g., consistently present in feces, non-pathogenic, easy to detect); name E. coli and Streptococcus faecalis or Coliform bacteria
• Part (e): Apply Manning's equation Q = (1/n) × A × R^(2/3) × S^(1/2) for full circular pipe; explain 150 mm minimum prevents clogging by solids and maintains self-cleansing velocity

Begin with a clear statement of given data and design criteria for both tanks in part (a). Allocate approximately 40% of effort to part (a) as it carries 20 marks, with detailed step-by-step calculations for rectangular and circular tank dimensions, followed by verification checks for retention time, horizontal velocity and weir overflow rate. For part (b) (15 marks), explain the theoretical basis for BOD/COD relationship, then enumerate practical reasons for divergence including biodegradability limitations, toxic substances, and microbial kinetics. For part (c) (15 marks), present the cropping pattern data clearly, apply sequential monthly water balance calculations accounting for crop water requirements, effective rainfall, canal losses (15%) and reservoir losses (10%) to determine storage capacity. Conclude with a summary table comparing both tank designs and key insights on BOD/COD differences.

• Part (a): Correct application of surface overflow rate formula (Q/As = 20 m/d) to determine plan areas; calculation of rectangular tank dimensions (L:B = 3:1, SWD = 3.5m) yielding L ≈ 55 m, B ≈ 18.3 m; circular tank diameter ≈ 35.7 m with SWD = 4.0 m
• Part (a): Verification of retention time (3-5 hours), horizontal velocity (<10 m/h), and weir overflow rate (6-10 m³/h/m) with clear pass/fail assessment; overall dimensions including inlet/outlet zones each equal to SWD
• Part (b): Explanation that COD measures total oxidizable matter while BOD measures only biodegradable fraction; reasons for BOD < COD including: presence of refractory organic compounds, toxic/inhibitory substances, insufficient acclimatization period, nitrification interference, and soluble non-biodegradable organics
• Part (b): Reference to typical BOD/COD ratios (0.4-0.8 for municipal wastewater, lower for industrial wastewater) and significance for treatment plant design
• Part (c): Monthly water balance computation: Crop water requirement (CWR) = CWR peak × kc × area; net irrigation requirement = CWR - effective rainfall; gross irrigation = net/(1-canal losses); reservoir yield accounting for 10% losses; storage capacity as maximum cumulative deficit
• Part (c): Application to typical Indian cropping pattern (e.g., kharif paddy + rabi wheat) with proper unit conversions (ha-m or Mm³) and final reservoir capacity determination

This is a multi-part numerical and descriptive problem requiring systematic solving. Allocate approximately 40% time to part (a) as it carries the highest marks (20) and involves comprehensive drain design using Lacey's regime theory; 30% each to parts (b) and (c). Begin with clear problem statements for each part, show all formulae with their sources, present step-by-step calculations with proper units, and conclude with practical significance of results.

• Part (a): Correct application of drainage coefficient to compute discharge Q = C×A = 0.06×(576×10⁴)/(24×3600) m³/s; application of Lacey's regime equations (P = 4.75√Q, R = 5V²/2f, S = f^(5/3)/(3340×Q^(1/6))) to determine trapezoidal section dimensions with 1:1 side slopes
• Part (b): Application of Thiem's equation for unconfined aquifer: Q = πK(h₂²-h₁²)/ln(r₂/r₁) to determine hydraulic conductivity K, then using same equation with r = well radius (0.25 m) to find drawdown in pumping well; or alternatively using Dupuit equation
• Part (c): Definition of grit as heavy inorganic solids (sand, gravel, silt, egg shells, coffee grounds) with particle size > 0.15 mm and specific gravity 2.4-2.65; explanation of grit removal necessity to prevent abrasion of pumps/mechanical equipment, pipe clogging, and reduction of digester volume
• Part (c): Explanation that velocity control devices (proportional flow weirs, Parshall flumes, venturi flumes, or sutro weirs) are essential in unaerated chambers to maintain 0.15-0.3 m/s velocity for grit settling while keeping organic matter in suspension; contrast with aerated chambers where air diffusion creates controlled rolling action making velocity devices redundant
• Integration: Recognition that Lacey's silt factor f=1.0 indicates medium silt typical of Indian alluvial plains, and that drainage design must account for monsoon intensity patterns in waterlogged areas like the Indo-Gangetic basin

Calculate the synthetic unit hydrograph coefficients Ct and Cp in part (a) using Snyder's method with given basin parameters; describe the sanitary landfill components with a neat schematic in part (b); and outline the EIA 2006 clearance process stepwise in part (c). Allocate approximately 40% time to numerical calculations in (a), 30% to diagram and description in (b), and 30% to procedural explanation in (c).

• Part (a): Correct application of Snyder's synthetic unit hydrograph equations to calculate Ct = tpR/(C₁(L.Lc)^0.3) and Cp = qpR.tR/(C₂.A)
• Part (a): Accurate substitution of values: L=125 km, Lc=90 km, A=3000 km², tR=12 h, tpR=34 h, qpR=135 m³/s.cm with proper unit handling
• Part (b): Description of sanitary landfill components: liner system, leachate collection, gas collection, cover system, monitoring wells
• Part (b): Schematic diagram showing cross-section with proper labeling of all functional layers and drainage systems
• Part (c): Sequential steps of EIA 2006: screening, scoping, public consultation, appraisal by EAC, final EC with conditions, post-clearance monitoring
• Part (c): Mention of category A/B projects, TOR preparation, 30-day public hearing notice, and validity period of clearance