Civil Engineering 2023 Paper I 50 marks Compulsory Solve

Q5

The flow rate of water over a weir is 3 m³/s. A 1 : 10 scale model of the weir is tested in a water channel. Answer the following : (i) What flow rate should be used for the model ? (ii) If a force of 15 N is experienced on the model, what force would be expected on the prototype ? 10 A rectangular wing on a small airplane has a 1·3 m chord and a 10 m span. When flying in air at 250 km/hour, the wing experiences a total aerodynamic force of 20 kN. If the lift to drag ratio is 3, what would be the lift coefficient of the wing ? Take density of air as 1·20 kg/m³. 10 An idealized radial turbine is rotating at 140 rev/min as shown in the figure. The absolute flow enters at 30° and leaves radially inward. The flow rate is 4·0 m³/s of water at 20°C. The blade thickness is constant at 10 cm. If density of water is 1000 kg/m³, what would be the theoretical power developed by the turbine ? 10 The shear stress induced at a depth of 7·0 m due to construction of a nearby foundation is 50 kN/m². The soil properties at the site are given below : Unit weight (γ) = 18 kN/m³ Effective cohesion (C') = 12 kN/m² Effective friction angle (φ') = 30° Compute the factor of safety against shear failure assuming water table located far below the point. Also compute the percentage reduction in factor of safety if water table rises to the ground level. Take unit weight of water = 9·81 kN/m³. 10 Excavation is made in a soil whose porosity is 35% and specific gravity of soil grains is 2·65. A 3·0 m layer of this soil is subjected to an upward seepage head of 4·0 m. What factor of safety exists against boiling (piping) ? If a factor of safety of 2 is required against boiling, what depth of gravel is required to be placed above the soil stratum ? Assume unit weight of gravel and the soil to be the same and loss of head in the layer to be negligible. Assume γw = 9·81 kN/m³.

हिंदी में प्रश्न पढ़ें

एक बीयर के ऊपर जल की प्रवाह दर 3 m³/s है । बीयर के एक 1 : 10 अनुमाप निदर्श का परीक्षण एक जल वाहिका में किया जाता है । निम्नलिखित के उत्तर दीजिए : (i) निदर्श के लिए किस प्रवाह दर का उपयोग किया जाना चाहिए ? (ii) आदिप्ररूप पर कितना बल प्रत्याशित होगा, यदि निदर्श पर 15 N के एक बल का अनुभव किया जाता है ? 10 एक छोटे हवाई जहाज पर एक आयताकार विमान पंख की जीवा 1·3 m और विस्तृति 10 m है । 250 km/hour पर वायु में उड़ने पर विमान पंख 20 kN के सकल वायुगतिक बल का अनुभव करता है । विमान पंख का उत्थान गुणांक क्या होगा, यदि उत्थान-विकर्ष अनुपात 3 है ? वायु का घनत्व 1·20 kg/m³ लीजिए । 10 चित्र में दर्शाए अनुसार, एक आदर्शीकृत त्रिज्यीय टरबाइन 140 परिक्रमण/मिनट पर घूर्णित है । निरपेक्ष प्रवाह 30° पर प्रवेश करता है और त्रिज्यीय अंतर्मुख निकलता है । 20°C पर जल की प्रवाह दर 4·0 m³/s है । ब्लेड की मोटाई 10 cm पर नियत है । टरबाइन द्वारा उत्पन्न की जाने वाली सैद्धांतिक शक्ति क्या होगी, यदि जल का घनत्व 1000 kg/m³ है । 10 निकट में एक नींव के निर्माण के कारण 7·0 m की गहराई पर उत्पन्न अपरूपण प्रतिबल 50 kN/m² है । स्थल पर मृदा गुण नीचे दिए गए हैं : एकक भार (γ) = 18 kN/m³ प्रभावी संसजन (C') = 12 kN/m² प्रभावी घर्षण कोण (φ') = 30° भौम जल स्तर को बिंदु से बहुत अधिक नीचे मानते हुए अपरूपण विफलता के विरुद्ध सुरक्षा गुणक की गणना कीजिए । सुरक्षा गुणक में प्रतिशत कमी की गणना भी कीजिए, यदि भौम जल स्तर भूमि तल तक आ जाए । जल का एकक भार = 9·81 kN/m³ लीजिए । 10 एक मृदा, जिसकी संरधता 35% और मृदा कणों का विशिष्ट घनत्व 2·65 है, में खुदाई की गई है । इस मृदा की एक 3·0 m परत पर 4·0 m की उपरिमुखी रिसन दाबोच्चता लगी है । क्वथन (पाइपिंग) के विरुद्ध सुरक्षा गुणक कितना है ? यदि क्वथन के विरुद्ध आवश्यक सुरक्षा गुणक 2 है, तो मृदा परत के ऊपर कितनी गहराई तक बजरी रखा जाना आवश्यक है ? बजरी और मृदा का एकक भार समान मान लीजिए और परत में दाबोच्चता में हानि नगण्य मान लीजिए । γw = 9·81 kN/m³ मान लीजिए ।

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Approach

Solve all five numerical problems systematically, allocating approximately 20% time to each part. Begin with dimensional analysis for the weir model (parts i-ii), then proceed to aerodynamic lift coefficient, turbine power calculation, factor of safety in shear failure, and finally seepage/boiling analysis. Present each solution with clear problem identification, formula application, substitution, and final answer with units.

Key points expected

  • Part (i-ii): Apply Froude's model law correctly — Qm/Qp = (Lr)^(5/2) = (1/10)^(5/2) = 0.00316, giving Qm = 9.49 L/s; Force ratio Fp/Fm = (Lr)^3 = 1000, giving Fp = 15,000 N
  • Part (iii): Calculate lift coefficient using CL = 2L/(ρV²A) where L = 15 kN (from 3:1 ratio), V = 69.44 m/s, A = 13 m²; yields CL ≈ 0.40
  • Part (iv): Apply Euler turbine equation P = ρQ(u1Vu1 - u2Vu2) with u2=0 (radial exit), u1 = ωr1, Vu1 = V1cos30°; requires geometric interpretation from figure for radius
  • Part (v): Compute σ = γz + Δσ = 18×7 + 50 = 176 kPa; τf = c' + (σ-u)tanφ' = 12 + 176×tan30° = 113.6 kPa; FS = τf/τ = 113.6/50 = 2.27; for WT at GL, FS reduces to ~1.14 (50% reduction)
  • Part (vi): Calculate critical hydraulic gradient ic = (Gs-1)/(1+e) = (2.65-1)/1.538 = 1.073; actual i = 4/3 = 1.333; FS = ic/i = 0.805 (unsafe); for FS=2, required gravel depth h gives total head loss 4m over (3+h)m with revised gradient

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies and applies: Froude's law for weir model scaling; lift-drag relationship and coefficient definition; Euler turbine equation with velocity triangle analysis; Mohr-Coulomb failure criterion with effective stress principle; critical hydraulic gradient and piping mechanismIdentifies most concepts correctly but confuses model laws (e.g., uses Reynolds instead of Froude) or misapplies effective vs total stress in shear strength calculationFundamental conceptual errors such as using geometric scaling for forces, ignoring lift-drag ratio, applying Bernoulli instead of Euler turbine equation, or using total stress for submerged conditions
Numerical accuracy20%10All calculations precise to 3 significant figures with correct unit conversions (km/h to m/s, rev/min to rad/s), proper handling of exponents in scaling laws, and accurate trigonometric evaluationsMinor arithmetic errors or unit conversion mistakes (e.g., missing 10³ factor in force scaling, incorrect velocity conversion) but methodologically soundMajor calculation errors, order-of-magnitude mistakes, or consistent unit confusion leading to unrealistic answers (e.g., model flow exceeding prototype)
Diagram quality15%7.5Clear velocity triangles for turbine inlet/outlet with 30° entry angle and radial exit; labeled stress elements for shear failure analysis; seepage flow net or head distribution for boiling problem; all diagrams dimensioned and annotatedBasic sketches present but missing critical labels or angles; turbine diagram lacks velocity components; no stress element shown for geotechnical partsNo diagrams despite problem requirements; or completely incorrect sketches that misrepresent flow patterns or stress conditions
Step-by-step derivation25%12.5Explicit statement of governing equations with physical meaning, clear substitution sequence, intermediate values shown, and dimensional verification at each step; derives void ratio from porosity before critical gradientCorrect final formulas but skips intermediate steps; assumes relationships without derivation (e.g., states scaling law without derivation from Froude number equality)No derivation shown — only final answers; or incorrect formula manipulation with missing steps that prevents verification
Practical interpretation20%10Interprets FS<1 in seepage as immediate piping risk requiring remedial gravel layer; discusses turbine power as theoretical maximum ignoring losses; comments on model-prototype validation for weir design; notes practical implications of water table rise on foundation safetyBrief mention of safety significance without elaboration; states answers are theoretical but lacks engineering contextNo physical interpretation of numerical results; fails to recognize that FS<1 indicates failure, or ignores practical relevance of model testing

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