Civil Engineering 2025 Paper I 50 marks Derive

Q6

(a) The discharge Q over a small rectangular weir depends on head H over the weir, the weir height P, gravity g, width of the weir L and fluid properties : density ρ, dynamic viscosity μ. Express the relationship between the variables in dimensionless form using Buckingham method. 15 marks (b) A rectangular channel is 4·0 m wide and carries a discharge of 5·0 m³/s at a depth of 1·0 m. A smooth contraction of the channel width is proposed at a section. Find the smallest contracted width that will not affect the upstream flow conditions. Neglect the energy losses in the transition. 15 marks (c) (i) In the given diagram, the height of a retaining wall is 5 m with a batter angle 6°. The back face of the retaining wall is supporting a sandy soil, the surface of which is sloping at an angle 12° with the horizontal. Determine the active force per unit length of the retaining wall. Also find the direction and location of the resultant force. The properties of backfill soil are as below : Angle of shearing resistance = 32° Cohesion = 0 Assume angle of wall friction as 50% of angle of shearing resistance. Bulk density = 15·5 kN/m³ 15 marks (ii) What is the effect of earthquake on lateral earth pressure against a retaining wall ? 5 marks

हिंदी में प्रश्न पढ़ें

(a) एक छोटे आयताकार विवर के ऊपर निस्सरण Q, विवर के ऊपर दाबोच्चता H, विवर की ऊँचाई P, गुरुत्व g, विवर की चौड़ाई L और तरल के गुणधर्मों : घनत्व ρ, गतिक श्यानता μ पर निर्भर करता है। बकिंघम विधि का उपयोग करते हुए चरों के बीच सम्बन्ध को विमारहित प्ररूप में व्यक्त कीजिए। 15 अंक (b) एक आयताकार वाहिका 4·0 m चौड़ी है और 1·0 m की गहराई पर 5·0 m³/s का निस्सरण वहन करती है । एक परिच्छेद पर वाहिका की चौड़ाई का मसृण संकुचन प्रस्तावित है । न्यूनतम संकुचित चौड़ाई ज्ञात कीजिए जिससे प्रतिप्रवाह प्रवाह अवस्थाएँ प्रभावित नहीं हों । संक्रमण में ऊर्जा ह्रास की उपेक्षा कीजिए । 15 अंक (c) (i) दिए गए आरेख में, 6° के आन्त कोण वाली प्रतिधारक भित्ति की ऊँचाई 5 m है । प्रतिधारक भित्ति का पश्च पृष्ठ रेतीली मृदा को आलम्ब प्रदान कर रहा है जिसकी सतह क्षैतिज से 12° के कोण पर आनत है । प्रतिधारक भित्ति की प्रति एकक लम्बाई पर सक्रिय बल को निर्धारित कीजिए । परिणामी बल की स्थिति और दिशा भी ज्ञात कीजिए । पश्च भराव मृदा के गुणधर्म इस प्रकार हैं : अपरूपण प्रतिरोध का कोण = 32° संसजन = 0 भित्ति के घर्षण कोण को अपरूपण प्रतिरोध के कोण का 50% मान लीजिए । स्थूल घनत्व = 15·5 kN/m³ 15 अंक (ii) एक प्रतिधारक भित्ति के विरुद्ध पार्श्व मृदा दाब पर भूकंप का क्या प्रभाव होता है ? 5 अंक

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Approach

Derive the dimensionless relationships for part (a) using Buckingham π-theorem with 7 variables and 3 fundamental dimensions, then solve the critical flow problem in part (b) by applying specific energy concepts and finding the minimum width for choked condition. For part (c)(i), calculate Coulomb active earth pressure with sloping backfill and wall batter using the wedge analysis, and for (c)(ii) explain seismic effects with Mononobe-Okabe approach. Allocate approximately 30% time to (a), 25% to (b), 35% to (c)(i), and 10% to (c)(ii).

Key points expected

  • For (a): Identify 7 variables (Q, H, P, g, L, ρ, μ), determine 3 fundamental dimensions (M, L, T), select 3 repeating variables (ρ, g, H), and derive 4 dimensionless π-terms including Reynolds number and Froude number variants
  • For (b): Calculate specific energy E = y + V²/2g = 1.319 m, establish critical depth condition at contraction, and solve for minimum contracted width b_min = 2.83 m using q_max = √(gE³/27)
  • For (c)(i): Apply Coulomb's active earth pressure theory with wall batter α = 6°, backfill slope β = 12°, φ = 32°, δ = 16°, compute K_a = 0.368, resultant force P_a = ½γH²K_a = 71.5 kN/m, and locate at H/3 from base inclined at θ = 16° to wall normal
  • For (c)(ii): Explain horizontal and vertical seismic coefficients, modified Mononobe-Okabe equation for seismic active earth pressure, and increased thrust typically 10-30% higher with shifted point of application
  • Correct handling of unit consistency throughout all calculations with proper SI units
  • Clear distinction between small weir (surface tension neglected) and large weir effects in dimensional analysis
  • Recognition that smooth contraction implies energy conservation and critical flow as limiting condition

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies all π-terms in (a) including Q/√(gH⁵) as dependent term; applies specific energy minimum principle for choked flow in (b); uses proper Coulomb wedge geometry with all three angles (α, β, φ) for inclined wall and sloping backfill in (c)(i); accurately describes Mononobe-Okabe seismic coefficients in (c)(ii)Identifies basic π-theorem steps but misses one π-term or makes minor dimensional errors; applies continuity and energy but confuses critical depth condition; uses Rankine instead of Coulomb or neglects wall batter angle; mentions earthquake increases pressure without coefficient explanationFundamental errors in Buckingham theorem application (wrong number of π-terms); confuses specific energy with total energy or uses Bernoulli incorrectly; applies hydrostatic pressure or uses wrong earth pressure theory; no knowledge of seismic effects on retaining walls
Numerical accuracy20%10All calculations precise: π-terms correctly formulated numerically; b_min = 2.83 m (or 2.8-2.9 m with reasonable rounding); K_a = 0.36-0.37, P_a = 70-73 kN/m with correct location at 1.67 m from base; seismic coefficient values realistic (k_h = 0.1-0.2 typical)Minor arithmetic errors in one part (e.g., π-term coefficients, specific energy calculation off by 5%, earth pressure coefficient within 10% of correct value); correct method but final answer slightly wrong; seismic effect mentioned without numerical illustrationMajor calculation errors in multiple parts; order of magnitude mistakes; completely wrong final answers despite correct method; no numerical work shown for (c)(ii)
Diagram quality15%7.5Clear labeled diagram for (a) showing weir geometry with H, P, L; specific energy curve for (b) showing E-y relationship with critical point; detailed wedge diagram for (c)(i) showing wall batter 6°, backfill slope 12°, failure plane, force polygon with W, P_a, R at proper angles; free body diagram for seismic forces in (c)(ii)Basic sketches present but missing key labels or dimensions; energy curve without critical point marked; wedge diagram without force vectors or angles; generic retaining wall without slope angles shownNo diagrams or incomprehensible sketches; diagrams contradict problem statement; missing essential elements like failure surface or pressure distribution
Step-by-step derivation25%12.5Complete systematic derivation: (a) explicit M-L-T matrix, repeating variables selection, algebraic solution for each π-term; (b) derivation of E = y + q²/2gy², differentiation for y_c, substitution to find q_max and b_min; (c)(i) full Coulomb derivation with trigonometric identity for K_a, force components, moment arm calculation; (c)(ii) derivation of seismic inertia forces and modified earth pressure equationKey steps shown but gaps in algebraic manipulation; jumps from energy equation to answer without showing critical condition derivation; uses standard formula for K_a without derivation; states seismic effect without showing force equilibriumNo derivation shown, only final formulas; incorrect derivation with conceptual errors; missing essential steps like dimensional matrix or critical flow condition; no explanation of how seismic forces modify static pressure
Practical interpretation20%10Interprets π-terms physically: Q/√(gH⁵) as discharge coefficient, Reynolds number for viscous effects, P/H and L/H as geometric similarity; explains practical significance of choked flow in canal design; discusses implications of wall batter on stability and construction; relates seismic design to Indian seismic zones (IS:1893) and importance of Mononobe-Okabe in Himalayan region retaining wallsBrief mention of physical meaning of dimensionless groups; notes that contraction below critical width causes upstream backwater; mentions wall batter affects pressure magnitude; states earthquakes are important for design without regional contextNo physical interpretation of results; purely mathematical treatment; no engineering significance discussed; ignores practical design implications entirely

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