Civil Engineering 2021 Paper I 50 marks Compulsory Solve

Q5

(a) A circular log of weight 1000 N and radius 18 cm is supported by a pair of brackets, one of which is shown in the figure below. Bar PN is hinged at 'P' and held by a string MN 67 cm long. To induce minimum tension at MN, determine the value of 2θ, as shown for equilibrium. Consider all contact surfaces smooth. Also find the value of minimum tension. (10 marks) (b) Determine the moment of resistance of the T-beam as shown in the figure. Use limit state method. Take M 20 grade of concrete and Fe 415 HYSD bars. (10 marks) (c) The fixed-end bar ABCD consists of three prismatic segments as shown in the figure. The end segments AB and CD have cross-sectional area 800 mm² and length 300 mm. The middle segment has cross-sectional area 1200 mm² and length 500 mm. Two axial loads P_B = 28·5 kN and P_C = 20 kN are acting in the bar as shown in the figure. Young modulus E is same for all three segments. (i) Determine the reaction forces at fixed supports. (ii) Determine the compressive axial force in the middle segment. (10 marks) (d) A flow meter when tested in a laboratory gives a pressure drop of 100 kN/m² for a discharge of 0·10 m³/s in a 150 mm diameter pipe. If a geometrically similar model is tested in 600 mm diameter pipe at identical conditions of fluid, determine the corresponding discharge and pressure drop in the model. (10 marks) (e) A sample of dry cohesionless soil whose angle of internal friction is 35°, is subjected to a triaxial test. If the minor principal stress (σ₃) is 105 kPa, at what values of deviator stress (Δσ) and major principal stress (σ₁) will the test specimen fail? (10 marks)

हिंदी में प्रश्न पढ़ें

(a) 1000 N भार और 18 cm त्रिज्या वाला एक वृत्ताकार लट्ठा, ब्रैकेट के एक जोड़े द्वारा आलंबित है जिनमें से एक नीचे चित्र में दर्शाया गया है । छड़ PN, 'P' पर हिंज है और 67 cm लंबी डोरी MN द्वारा बंधी है । MN में न्यूनतम तनाव उत्पन्न करने के लिए, दर्शाई गई साम्यावस्था के लिए, 2θ के मान को निर्धारित कीजिए । सभी संपर्क सतहों को चिकना मान लीजिए । न्यूनतम तनाव का मान भी ज्ञात कीजिए । (10 अंक) (b) चित्र में दर्शाई गई T-धन के आघूर्ण प्रतिरोध का निर्धारण कीजिए । सीमंत अवस्था विधि का उपयोग कीजिए । M 20 ग्रेड की कंक्रीट और Fe 415 एच.वाई.एस.डी. छड़ें लीजिए । (10 अंक) (c) चित्र में दर्शाए अनुसार आबद्ध-सिरा छड़ ABCD तीन समपार्श्वीय खंडों से बनी है । अंत खंडों AB और CD का अनुप्रस्थ-परिच्छेद क्षेत्रफल 800 mm² और लंबाई 300 mm है । मध्य खंड का अनुप्रस्थ-परिच्छेद क्षेत्रफल 1200 mm² और लंबाई 500 mm है । दो अक्षीय भार P_B = 28·5 kN और P_C = 20 kN चित्र में दर्शाए अनुसार छड़ में लगे हैं । यांग मापांक E सभी तीन खंडों के लिए समान है । (i) आबद्ध आलम्बों पर प्रतिक्रिया बलों का निर्धारण कीजिए । (ii) मध्य खंड में संपीडन अक्षीय बल का निर्धारण कीजिए । (10 अंक) (d) एक प्रवाह मापी, एक प्रयोगशाला में परीक्षण करने पर, एक 150 mm व्यास की पाइप में एक 0·10 m³/s के निस्सरण के लिए 100 kN/m² का दाब पात देता है । तरल की समान अवस्था पर यदि एक ज्यामितियतः समरूप निदर्श का परीक्षण 600 mm व्यास की पाइप में किया जाता है, तो निदर्श में संगत निस्सरण और दाब पात का निर्धारण कीजिए । (10 अंक) (e) एक शुष्क संसजनहीन मृदा, जिसका आंतरिक घर्षण कोण 35° है, के प्रतिदर्श पर एक त्रिअक्षीय परीक्षण किया जाता है । विचलक प्रतिबल (Δσ) और उच्च मुख्य प्रतिबल (σ₁) के किन मानों पर परीक्षण प्रतिदर्श भंग होगा, यदि निम्न मुख्य प्रतिबल (σ₃) 105 kPa है ? (10 अंक)

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Approach

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.

Key points expected

  • 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

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies equilibrium conditions for (a), limit state assumptions for (b), compatibility equations for (c), appropriate similarity laws for (d), and Mohr-Coulomb failure theory for (e); no conceptual errors in any sub-partMinor errors in one sub-part such as wrong similarity law for (d) or incorrect failure criterion for (e); core concepts mostly sound but some confusion between working stress and limit state methodsFundamental misconceptions like using static determinacy for indeterminate bar (c), applying Bernoulli instead of dimensional analysis for (d), or using Rankine's theory instead of Mohr-Coulomb for (e)
Numerical accuracy20%10All calculations precise to 3 significant figures; correct unit conversions (cm to m, kPa to N/mm²); final answers properly boxed with units; no arithmetic errors in any sub-partMinor calculation errors in 1-2 sub-parts (e.g., wrong power in similarity scaling, arithmetic slip in solving simultaneous equations); correct methodology but final answers slightly offMajor numerical errors in multiple sub-parts; inconsistent units (mixing cm and m); order-of-magnitude mistakes; no unit specification in final answers
Diagram quality20%10Clear free-body diagrams for (a), (c), (e) with all forces labeled; T-beam cross-section for (b) showing neutral axis, stress blocks; flow meter sketch for (d); Mohr's circle or stress element for (e); diagrams aid solution directlyDiagrams present but incomplete—missing some force components, dimensions not labeled, or rough sketches without proper proportion; diagrams don't fully support the derivationNo diagrams or irrelevant sketches; failure to visualize the physical problem; missing critical diagrams like FBD for three-force system in (a) or deformed shape for (c)
Step-by-step derivation20%10Logical progression from given data to final answer for each sub-part; explicit statement of equations used (ΣM=0, ε_total=0, etc.); clear algebraic manipulation; intermediate results shown; cross-checks included where applicableSteps present but some jumps in logic; missing explicit statement of key equations; final answer appears without clear derivation path; one sub-part may lack sufficient workingDisorganized working with no clear structure; jumps directly to answers; missing essential steps like compatibility condition in (c) or derivation of scaling exponents in (d); illegible or cramped presentation
Practical interpretation20%10Interprets minimum tension condition in (a) as optimal bracket design; relates T-beam moment capacity to IS code safety; discusses why middle segment compresses in (c); explains model-prototype scaling for dam/spillway design in India; connects triaxial results to slope stabilityBrief mention of practical relevance without elaboration; standard concluding statements without specific engineering context; misses opportunity to relate to Indian infrastructure examplesPurely mathematical treatment with no physical interpretation; no discussion of why results matter; fails to connect to civil engineering practice or code provisions

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