Civil Engineering 2025 Paper I 50 marks Solve

Q2

(a) A T-section beam is constructed by gluing two pieces of wood together as shown in the figure. The maximum stress in the glue joints is to be limited to 2 MPa in tension and maximum shear stress is to be limited to 1·7 MPa. (i) Determine the stress components on element at point 'P'. Point 'P' is located at glued joint. (ii) Determine principal stresses at point 'P'. (iii) Show these stresses on properly oriented 2-D elements. (iv) Determine the maximum value for load 'w'. 20 marks (b) For the beam shown in the figure, find the reaction at C and draw the bending moment diagram for the beam. Take EI = Constant. 10 marks (c) For the column section shown in the figure, determine the design strength components corresponding to the condition of 'balanced failure'. Assume M25 grade concrete and Fe 500 grade steel. Consider loading eccentricity with respect to the major axis alone. Assume 8 φ ties and 40 mm clear cover. Take Es = 2 × 10⁵ N/mm². For Fe 500 steel εy = (0.87 fy/Es) + 0.002. Cc = resultant force in concrete. Cs = Σ(i=1 to n) (fsi - fci) Asi. Mc = Cc(D/2 - X̄). Ms = Σ(i=1 to n) (fsi - fci) Asi × yi. Asi → Area of steel in ith row. yi → Distance of ith row of steel from the centroidal axis. Design stress at specified strains for Fe 500: Strain 0.000 → Stress 0.0 MPa; 0.00174 → 347.8 MPa; 0.00195 → 369.6 MPa; 0.00226 → 391.3 MPa; 0.00277 → 413.0 MPa; 0.00312 → 423.9 MPa; 0.00417 → 434.8 MPa. 20 marks

हिंदी में प्रश्न पढ़ें

(a) एक T-परिछेद धरन का निर्माण, चित्र में दर्शाए अनुसार, लकड़ी के दो टुकड़ों को चिपकाकर किया गया है। गोंद जोड़ में अधिकतम प्रतिबल, तनन में 2 MPa तक सीमित किया जाना है और अधिकतम अपरूपण प्रतिबल 1·7 MPa तक सीमित किया जाना है। (i) अवयव के बिंदु 'P' पर प्रतिबल घटकों को निर्धारित कीजिए। बिंदु 'P' चिपकाए हुए जोड़ पर स्थित है। (ii) बिंदु 'P' पर मुख्य प्रतिबल निर्धारित कीजिए। (iii) इन प्रतिबलों को उचित रूप से अनुमुख द्वि-आयामी अवयवों पर दर्शाइए। (iv) भार 'w' के अधिकतम मान को निर्धारित कीजिए। 20 अंक (b) चित्र में दर्शाई गई धरन के लिए, C पर प्रतिक्रिया ज्ञात कीजिए और धरन के लिए बक्रन आघूर्ण आरेख बनाइए। EI = नियत लीजिए। 10 अंक (c) चित्र में दर्शाए गए स्तंभ परिच्छेद के लिए, 'संतुलित विफलन' अवस्था के अनुरूप अभिकल्पन सामर्थ्य घटकों को निर्धारित कीजिए। M25 ग्रेड कंक्रीट और Fe 500 ग्रेड इस्पात मान लीजिए। भारण उत्केंद्रता केवल दीर्घ अक्ष के सापेक्ष लीजिए। 8 φ के बंधक और 40 mm का स्पष्ट आवरण मान लीजिए। Es = 2 × 10⁵ N/mm² लीजिए। Fe 500 इस्पात के लिए εy = (0.87 fy/Es) + 0.002। Cc = कंक्रीट में परिणामी बल। Cs = Σ(i=1 से n) (fsi - fci) Asi। Mc = Cc(D/2 - X̄)। Ms = Σ(i=1 से n) (fsi - fci) Asi × yi। Asi → इवी पंक्ति में इस्पात का क्षेत्रफल। yi → इस्पात की इवी पंक्ति की केंद्रक अक्ष से दूरी। Fe 500 के लिए निर्दिष्ट विकृतियों पर अभिकल्पन प्रतिबल: विकृति 0.000 → प्रतिबल 0.0 MPa; 0.00174 → 347.8 MPa; 0.00195 → 369.6 MPa; 0.00226 → 391.3 MPa; 0.00277 → 413.0 MPa; 0.00312 → 423.9 MPa; 0.00417 → 434.8 MPa। 20 अंक

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Approach

Solve this multi-part structural analysis problem by allocating approximately 40% time to part (a) given its 20 marks weightage, 20% to part (b), and 40% to part (c). Begin with calculating section properties and stress transformations for the T-beam glue joint in (a), then apply compatibility/ equilibrium methods for the indeterminate beam in (b), and finally perform strain-compatibility analysis for the reinforced concrete column balanced failure condition in (c). Present all derivations systematically with clear free-body diagrams and stress element sketches.

Key points expected

  • For (a)(i)-(iv): Calculate centroid and moment of inertia of T-section, determine bending and shear stress components at glued joint P, apply stress transformation equations for principal stresses, sketch Mohr's circle or rotated stress elements, and establish limiting criteria for maximum distributed load w based on glue joint capacity (2 MPa tension, 1.7 MPa shear)
  • For (b): Apply force method or moment-area method to solve statically indeterminate beam, establish compatibility equation for support settlement or rotation, calculate reaction at C, and construct complete bending moment diagram showing salient values at supports and midspan
  • For (c): Determine balanced failure condition where concrete reaches 0.0035 strain simultaneously with steel reaching yield strain (0.87fy/Es + 0.002), locate neutral axis depth, calculate Cc using stress block parameters for M25 concrete, determine Cs by summing contributions from all steel rows using given design stress-strain data for Fe 500, and compute Mc and Ms to obtain ultimate moment capacity
  • Correct application of IS 456:2000 provisions for stress block parameters (xumax/d = 0.46 for Fe 500) and clear cover requirements for 40mm cover with 8φ ties
  • Proper use of transformed section properties and parallel axis theorem for composite T-section analysis in timber beam design
  • Accurate interpolation from provided Fe 500 design stress table for intermediate strain values in column analysis
  • Clear presentation of principal stress orientation angles and maximum shear stress planes on properly oriented 2-D stress elements

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies that (a) requires combined bending-shear analysis at glue joint with principal stress transformation; (b) requires indeterminate beam analysis using consistent deformation method; (c) applies strain compatibility for balanced failure with correct limiting strains (0.0035 concrete, 0.00226+ for Fe 500 steel) and IS 456 stress block parametersIdentifies basic concepts but confuses pure bending with combined loading in (a), uses incorrect degree of indeterminacy in (b), or misapplies balanced failure condition by ignoring concrete strain limit or using incorrect steel yield strain formula in (c)Fundamental conceptual errors such as treating glued joint as pure shear only, applying simple support reactions to indeterminate beam, or using working stress method instead of limit state for column design
Numerical accuracy25%12.5All calculations precise to 3 significant figures: correct section modulus (I/ymax) for T-section, accurate principal stress magnitudes and angles using σ1,2 = (σx+σy)/2 ± √[((σx-σy)/2)²+τxy²], correct reaction at C within 2% tolerance, and balanced failure Pu, Mu matching IS 456 design values with proper interpolation from given stress tableMinor arithmetic errors in section properties or principal stress calculation; reaction at C within 10% tolerance; column capacity calculations with incorrect neutral axis depth but correct methodologyMajor calculation errors: wrong moment of inertia by factor >2, incorrect principal stress formula application, reaction magnitude wrong by >20%, or completely unrealistic column capacity values violating strain compatibility
Diagram quality15%7.5Clear dimensioned sketches: T-section with centroid location and neutral axis for (a); free-body diagram with deflected shape for indeterminate beam in (b); column cross-section with strain distribution, stress block, and force resultants (Cc, Cs) for (c); plus properly oriented stress elements showing σ1, σ2, τmax with correct rotation angles for (a)(iii)Diagrams present but missing key dimensions or labels; stress elements shown without proper orientation angles; bending moment diagram missing values at critical sectionsMissing essential diagrams, unrecognizable sketches, or diagrams contradicting calculations (e.g., tension side shown incorrectly, stress block drawn above neutral axis)
Step-by-step derivation25%12.5Systematic derivation showing: (a) section property calculation → stress components → Mohr's circle or transformation equations → principal stresses → load limit determination; (b) choice of redundant → compatibility equation setup → solution for reaction → BMD construction; (c) assumption of strains → neutral axis location → stress resultants → equilibrium verification → moment capacity with all Σ terms expandedDerivations present but with skipped steps (e.g., direct formula application without showing stress transformation, compatibility equation stated without derivation); final answers correct but logic gaps visibleNo derivations shown—only final answers stated; or incorrect derivation sequence (e.g., assuming load before checking stress limits, applying moment-area without establishing baseline)
Practical interpretation15%7.5Interprets (a) result as governing criterion for timber joint design with explicit comparison of tension vs shear limits; explains (b) reaction magnitude in context of relative stiffness and load sharing; for (c) identifies balanced failure as transition between tension and compression-controlled sections per IS 456, noting practical significance for ductility and failure mode prediction in RC columnsBrief mention of practical relevance without elaboration; or generic statements about safety without specific connection to calculated valuesNo practical interpretation provided; or completely incorrect interpretation (e.g., stating balanced failure is undesirable when it actually defines optimal section utilization, confusing glue shear with beam shear failure mode)

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