Mechanical Engineering 2024 Paper I 50 marks Solve

Q3

(a) Show the loading on the beams corresponding to the bending moment diagrams shown below. The beams are simply supported at A and B : (20 marks) (b) A rectangular plate with a central circular hole of diameter 250 mm is subjected to two mutually perpendicular direct stresses of magnitudes 20 MN/m² and 10 MN/m² along the x and y directions, respectively. In addition, a shear stress of magnitude 7·5 MN/m² acts on all the four planes on which the direct stresses act (see the figure below). The circular hole is deformed into the shape of an ellipse. If the value of Young's modulus of the material is 210 GN/m² and Poisson's ratio is 0·3, find the lengths of the major and minor axes : (20 marks) (c) Compare different types of cast iron in respect of (i) form of carbon, (ii) micro-structure and (iii) mechanical properties. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) दर्शाई गई धरनों के लिए भार ज्ञात कीजिए, जिनके बंकन आधुर्ण आरेख नीचे दिए गए हैं। टेक A और B पर धरने शुडालंबित हैं : (20 अंक) (b) एक आयताकार प्लेट के बीच में 250 mm व्यास का एक गोलाकार छेद है। इस प्लेट पर x और y दिशाओं में क्रमशः 20 MN/m² तथा 10 MN/m² परिमाण के दो परस्पर लंबवत् प्रत्यक्ष प्रतिबल लग रहे हैं। इसके अलावा, 7·5 MN/m² परिमाण का अपरूपण प्रतिबल उन सभी चार सतहों पर कार्य करता है, जिन पर प्रत्यक्ष प्रतिबल कार्य करता है (नीचे दिए गए चित्र को देखिए)। गोलाकार छेद दीर्घवृत्त के आकार में विकृत हो जाता है। यदि पदार्थ का यंग मापांक 210 GN/m² तथा प्वासों अनुपात 0·3 है, तो दीर्घ और लघु अक्षों की लंबाइयाँ ज्ञात कीजिए : (20 अंक) (c) विभिन्न प्रकार के ढलवां लोहे की (i) कार्बन के प्ररूप, (ii) सूक्ष्म संरचना तथा (iii) यांत्रिक गुणों के आधार पर तुलना कीजिए। (10 अंक)

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Approach

Solve part (a) by reverse-engineering loads from given BMD shapes using d²M/dx² = -w and boundary conditions for simply supported beams. For part (b), apply stress concentration theory for elliptical holes under biaxial loading with shear, using Kirsch's solution and strain transformation to find deformed axes. For part (c), construct a comparative table covering gray, white, malleable, and spheroidal graphite cast irons. Allocate approximately 35% time to (a), 40% to (b), and 25% to (c) based on marks distribution.

Key points expected

  • Part (a): Reverse BMD analysis—triangular BMD implies concentrated moment at midspan, parabolic BMD implies UDL, trapezoidal BMD implies combined loading; apply dV/dx = -w and dM/dx = V relationships
  • Part (a): Correct identification of load cases—point loads, UDL, moments, or combinations that produce given BMD shapes with zero moments at supports A and B
  • Part (b): Stress transformation to principal stresses: σ₁,₂ = (σₓ+σᵧ)/2 ± √[((σₓ-σᵧ)/2)² + τₓᵧ²] = 15 ± 9.01 = 24.01 and 5.99 MN/m²
  • Part (b): Stress concentration at hole boundary using Kirsch solution; tangential stress σθ = σ₁(1+2cos2θ) + σ₂(1-2cos2θ) at r = a
  • Part (b): Ellipse deformation calculation using strains ε₁ = (σ₁-νσ₂)/E and ε₂ = (σ₂-νσ₁)/E; major axis = d(1+ε₁), minor axis = d(1+ε₂)
  • Part (c): Gray CI—flake graphite, ferrite/pearlite matrix, good damping, poor toughness; White CI—cementite, pearlite/ledeburite, hard, brittle; Malleable CI—temper carbon nodules, ferrite/pearlite, better toughness; SG/ductile iron—spheroidal graphite, ferrite/pearlite, best strength-toughness combination
  • Part (c): Microstructural comparison with sketches of graphite morphology; mechanical properties table with tensile strength, % elongation, hardness values

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness22%11Part (a) correctly applies differential relationships between load, shear and moment; part (b) properly uses plane stress transformation, Kirsch stress concentration for elliptical holes, and strain-stress relationships; part (c) accurately distinguishes all four cast iron types by carbon form, microstructure and properties with correct metallurgical terminologyCorrect basic concepts in each part but minor errors—e.g., misses shear stress contribution in principal stress calculation, or confuses malleable with SG iron microstructuresFundamental misconceptions—treats BMD slopes incorrectly, ignores stress concentration factor for hole, or groups all cast irons as 'brittle' without distinction
Numerical accuracy20%10Part (b): Principal stresses σ₁=24.01 MN/m², σ₂=5.99 MN/m²; strains ε₁=1.028×10⁻⁴, ε₂=9.52×10⁻⁶; major axis=250.0257 mm, minor axis=250.0024 mm; all calculations show 3-4 significant figures with proper unit handling (MN/m² to GN/m² conversion)Correct methodology but arithmetic slips—e.g., principal stress calculation error, or Poisson's ratio applied incorrectly in one term; final axes within 5% of correct valuesMajor numerical errors—wrong formula for principal stresses, ignores E or ν, or order-of-magnitude errors in final dimensions (mm vs m confusion)
Diagram quality18%9Part (a): Clear loading diagrams with forces/moments labelled, dimensions shown, and correspondence to given BMDs explicitly indicated; part (c): Microstructural sketches showing flake, cementite, temper carbon nodules, and spheroidal graphite morphologies with proper labellingDiagrams present but incomplete—loading diagrams lack dimensions or arrows, or cast iron microstructures drawn without clear graphite shape distinctionMissing or misleading diagrams—no loading diagrams for (a), or generic iron-carbon phase diagram instead of specific cast iron microstructures for (c)
Step-by-step derivation22%11Part (a): Shows complete reverse analysis—states d²M/dx² = -w, integrates to find V and w, checks boundary conditions; part (b): Explicit stress transformation equations, Mohr's circle or direct substitution shown, Kirsch solution stated, strain calculation with Hooke's law for plane stress, final dimension calculation; part (c): Structured comparison with clear criteria headersDerivations present but abbreviated—jumps from given data to principal stresses without showing τ contribution, or states Kirsch result without context; comparison lacks systematic structureNo derivations—final answers stated without working; or incorrect derivation sequence (e.g., calculates strains before finding principal stresses)
Practical interpretation18%9Part (a): Comments on structural identification from BMDs—diagnostic use in forensic engineering; part (b): Discusses significance of small deformations (0.01% order) for precision components, notes stress concentration factor importance for fatigue life; part (c): Links cast iron selection to applications—gray CI for machine beds (damping), SG iron for automotive components, white CI for wear resistanceBrief mention of applications without specific examples; notes deformation is 'small' without quantitative contextNo interpretation—purely mathematical/nominal treatment; or incorrect applications (e.g., suggests white CI for structural beams)

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