Q8
(a) At a site, fine sand exists to a depth of 10 m and below this lies a soft clay layer 7·0 m thick. Water table is 4·0 m below the ground surface. Saturated unit weight of sand is 20·0 kN/m³ and the wet unit weight above the water table is 18 kN/m³. The water content of the normally consolidated clay is 42%, liquid limit is 46% and the specific gravity of the solid particles is 2·75. The proposed construction will transmit a net stress of 130 kN/m² at the centre of the clay layer. Find the average settlement of the clay layer. (15 marks) (b) A strip footing of width 2·8 m as shown in the figure is founded at a depth of 2·5 m below the ground surface in a C – φ soil. Water table is at a depth of 6 m below the ground surface. The average moist weight of soil above the water table is 18 kN/m³. Determine the ultimate bearing capacity, net ultimate bearing capacity, net allowable bearing pressure and the load/m for a factor of safety of 2·5. Use the general shear failure theory of Terzaghi. Given : For φ = 30°, Nc = 37·2 Nq = 22·5 Nγ = 19·7 What will be the percent decrease in ultimate bearing capacity if during the flooding, water level rises 2 m above around surface ? (15 marks) (c) Water at 20°C flows through a pipe of inlet diameter of 10 cm and passes further through a circular nozzle of diameter 2·5 cm, exits into the air as a jet, and strikes a vertical plate as shown in the figure. A force, F = 100 N is required to hold the plate stationary. Assuming steady, frictionless, one-dimensional flow and densities of water and mercury as 1000 kg/m³ and 13550 kg/m³ respectively, answer the following : (i) Determine the velocities at sections ① and ②. (ii) Determine the mass flow rate of water. (iii) Determine the mercury manometer reading 'h'. (20 marks)
हिंदी में प्रश्न पढ़ें
(a) एक स्थल पर, 10 m गहराई तक महीन बालू है और इसके नीचे 7·0 m मोटी मृद् मृत्तिका परत है । भौम जल स्तर भूमि तल से 4·0 m नीचे है । बालू का संतृप्त एकक भार 20·0 kN/m³ है और भौम जल स्तर से ऊपर आर्द्र एकक भार 18 kN/m³ है । सामान्य रूप से संघनित मृत्तिका का जलांश 42%, द्रव सीमा 46% है और ठोस कणों का विशिष्ट घनत्व 2·75 है । प्रस्तावित निर्माण, मृत्तिका परत के मध्य पर 130 kN/m² का निवल प्रतिबल प्रेषित करेगा । मृत्तिका परत का औसत निष्पदन ज्ञात कीजिए । (15 marks) (b) 2.8 m चौड़ी एक पट्टी पाद (फुटिंग) को चित्र में दर्शाए अनुसार एक C - φ मृदा में भूमि तल से 2.5 m नीचे आधारित किया गया है। भौम जल स्तर भूमि तल से नीचे 6 m गहराई पर है। भौम जल स्तर से ऊपर मृदा का औसत आर्द्र भार 18 kN/m³ है। 2.5 के एक सुरक्षा गुणक के लिए चरम धारण क्षमता, निवल चरम धारण क्षमता, निवल अनुमेय धारण दाब और भार/मीटर निर्धारित कीजिए। टेरज़ाघी के सामान्य अपरूपण विफलन सिद्धांत का उपयोग कीजिए। प्रदत्त : φ = 30° के लिए, Nc = 37.2 Nq = 22.5 Nγ = 19.7 यदि बाढ़ के दौरान, जल स्तर भूमि तल से 2 m ऊपर हो जाता है तो चरम धारण क्षमता में प्रतिशत कमी क्या होगी ? पाद (फुटिंग) ←B = 2.8 m→ φ = 30° γ = 18 kN/m³ C = 40 kN/m² (15 marks) (c) 10 cm के अंतर्गम व्यास के एक पाइप से 20°C पर जल प्रवाहित है और आगे यह 2·5 cm व्यास के एक वृत्ताकार नोजल से गुजरते हुए वायु में एक जेट की तरह बाहर निकलता है और चित्र में दर्शाए अनुसार एक उर्ध्वाधर प्लेट से टकराता है। प्लेट को स्थिर रखने के लिए एक बल, F = 100 N आवश्यक है। प्रवाह को अपरिवर्ती, घर्षण रहित, एक-विमीय प्रवाह और जल तथा पारे के घनत्व क्रमशः: 1000 kg/m³ और 13550 kg/m³ मानते हुए, निम्नलिखित के उत्तर दीजिए : (i) परिच्छेद ① और ② पर वेग निर्धारित कीजिए। (ii) जल की द्रव्यमान प्रवाह दर निर्धारित कीजिए। (iii) पारा मैनोमीटर पाठ्यांक 'h' निर्धारित कीजिए। (20 marks)
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How this answer will be evaluated
Approach
Solve all three parts systematically, allocating approximately 30% time to part (a) settlement calculation, 30% to part (b) bearing capacity with Terzaghi's theory, and 40% to part (c) fluid mechanics with continuity, momentum and manometry. Begin each part with clear identification of given data, apply relevant formulas with proper unit conversions, and conclude with boxed final answers. For part (c), solve sub-parts (i)-(iii) sequentially as they are interdependent.
Key points expected
- Part (a): Calculate initial effective stress at mid-clay layer using submerged unit weights; determine compression index Cc from liquid limit using Cc = 0.009(LL-10); compute settlement using ΔH = (CcH₀/1+e₀)log₁₀[(σ'₀+Δσ)/σ'₀]
- Part (b): Apply Terzaghi's general shear failure equation for strip footing: qu = cNc + γDfNq + 0.5γBNγ; calculate net ultimate and allowable bearing pressures; determine percent decrease when water table rises to ground surface using submerged unit weight
- Part (c)(i): Apply momentum equation F = ρQ(V₂-0) = ρA₂V₂² to find V₂, then continuity equation A₁V₁ = A₂V₂ to find V₁
- Part (c)(ii): Calculate mass flow rate ṁ = ρA₁V₁ = ρA₂V₂ using velocity from (i)
- Part (c)(iii): Apply Bernoulli's equation between sections ① and ② including manometer reading h with mercury-water interface; solve for h using pressure balance ρw(V₁²-V₂²)/2 = (ρm-ρw)gh
- For all parts: Show proper unit conversions (kN/m³ to kg/m³ where needed), use g = 9.81 m/s², and state all assumptions clearly
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Concept correctness | 25% | 12.5 | Correctly identifies and applies: (a) Skempton's compression index correlation for normally consolidated clay with proper void ratio calculation; (b) Terzaghi's bearing capacity factors with correct shape/depth factors and water table correction; (c) momentum principle for jet impact, continuity equation, and manometry with proper pressure balance at mercury-water interface | Uses correct basic formulas but makes minor errors in identifying normally consolidated clay behavior, misapplies water table correction factor, or confuses absolute/gauge pressure in manometer equation | Uses wrong consolidation theory (e.g., overconsolidated), applies wrong bearing capacity theory (general vs. local shear), or fundamentally misunderstands momentum transfer in fluid jet |
| Numerical accuracy | 25% | 12.5 | All calculations precise to 3 significant figures: (a) settlement ~130-150 mm range; (b) qu ~800-900 kN/m² with net allowable ~300-350 kN/m², percent decrease ~45-50%; (c) V₁ ~0.8 m/s, V₂ ~12.7 m/s, ṁ ~6.3 kg/s, h ~0.78 m; proper handling of unit conversions throughout | Correct method but arithmetic errors in final values (±15% tolerance), or inconsistent use of g = 10 vs 9.81 m/s², or minor errors in area/diameter conversions | Order of magnitude errors, wrong unit conversions (cm to m), or calculation errors exceeding ±25% indicating fundamental misunderstanding |
| Diagram quality | 15% | 7.5 | Clear soil profile diagram for (a) showing layers, water table, stress distribution; footing sketch for (b) with dimensions, failure surface; detailed flow diagram for (c) with sections ①②, nozzle, plate, manometer connection with proper labeling of h | Basic sketches present but missing key labels (e.g., no failure surface in (b), no manometer detail in (c)), or diagrams too small/cluttered | No diagrams despite figure references in question, or diagrams completely misrepresenting the physical setup (e.g., wrong flow direction, missing water table) |
| Step-by-step derivation | 20% | 10 | Every formula stated before substitution, clear substitution sequence: (a) γsat → γsub → σ'₀ → e₀ → Cc → ΔH; (b) identify c=0 for sand, apply Terzaghi equation stepwise, then water table correction; (c) momentum → V₂ → continuity → V₁ → mass flow → Bernoulli → manometer | Correct final formulas shown but skips intermediate steps or combines calculations, making error tracing difficult; some substitution steps implied rather than explicit | Jumps directly to final answers without showing formulas, or presents disconnected calculations without logical flow between steps |
| Practical interpretation | 15% | 7.5 | Interprets settlement magnitude for structure safety in (a); discusses bearing capacity implications for foundation design and flood risk in (b); relates manometer reading to practical flow measurement and force magnitude in (c); notes assumptions limiting real-world accuracy | Brief concluding statements without substantive interpretation, or generic comments not specific to calculated values | No interpretation of results, or physically absurd conclusions (e.g., settlement acceptable when >500mm, no mention of flooding risk for foundations) |
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