(a) An aluminium tensile specimen has a diameter of 30·50 mm and a gauge length 275 mm. If the force of 17·50 × 10⁴ N elongates the gauge length by 1·28 mm, determine the Poisson's ratio and the modulus of elasticity. Also, determine by how much the

Question

(a) An aluminium tensile specimen has a diameter of 30·50 mm and a gauge length 275 mm. If the force of 17·50 × 10⁴ N elongates the gauge length by 1·28 mm, determine the Poisson's ratio and the modulus of elasticity. Also, determine by how much the force causes the diameter of the specimen to contract. Assume shear modulus G = 22 GPa and yield strength σᵧ = 435 N/mm². (10 marks)

(b) A solid steel shaft of diameter 65 mm is to be designed using an allowable shear stress τₐₗₗₒw = 60 N/mm² and an allowable angle of twist per unit length θ = 1·05° per metre. Determine the maximum permissible torque that may be applied to the shaft. Take shear modulus as 80 GPa. (10 marks)

(c) A rigid box of mass 85 kg shown in the figure below rests on a floor. The coefficient of static friction for the contact surface is 0·25. What will be the maximum force, 'F' and the highest position, 'h' of its application so that the rigid box neither slides on the floor nor tips over? (10 marks)

(d) As shown in the figure, a beam of symmetrical I-section spanning 8·0 m is prestressed by a parabolic cable with an eccentricity of 150 mm at the centre of the span and zero at supports. The beam supports a uniformly distributed live load of 2·5 kN/m.

(i) Find the effective force in the cable for balancing the dead and live loads on the beam.

(ii) Calculate the shift of the pressure line from the tendon's centre line.

Take unit weight of concrete as 24 kN/m³.
(All dimensions are in mm) (10 marks)

(e) A tie member consisting of an ISA 75 × 50 × 8 (E 250 grade of steel) is connected to a 12 mm thick gusset plate using a 6 mm fillet weld at site. The welding is done on its three sides as shown in the figure. The angle between fusion faces is 75°. Find the lengths of weld L_w₁ and L_w₂, if the connection is designed to transmit a load equal to the design strength of the member. (10 marks)

For ISA 75 × 50 × 8, A_g = 938 mm² and C_xx = 25·2 mm

Take γ_mo = 1·10 and for site welding, γ_mw = 1·5.

K = 0·7 for 60° – 90° angle between fusion faces.

For E 250 grade steel :
f_u = 410 MPa
f_y = 250 MPa

UPSC Answer Check · Accepted Answer

Calculate the required values for all five parts systematically, spending approximately 15% time on (a), 15% on (b), 20% on (c), 35% on (d) as it has two sub-parts with prestressing calculations, and 15% on (e). Begin each part with stating the relevant formula, show substitution with units, compute step-by-step, and conclude with the final answer and appropriate units. For parts (c) and (d) requiring figures, sketch clear free-body diagrams and cable profiles respectively.
- Part (a): Calculate longitudinal strain, lateral strain using G = E/[2(1+ν)] relationship, then find Poisson's ratio ν, modulus of elasticity E, and diameter contraction using correct sign convention
- Part (b): Determine maximum permissible torque by checking both shear stress criterion (τ = 16T/πd³) and angle of twist criterion (θ = T/GJ in rad/m), then select the governing lower value
- Part (c): Draw FBD showing weight, applied force F at height h, normal reactions, and friction; establish equilibrium equations for sliding condition (F = μW) and tipping condition (moment about leading edge), solve simultaneously for F_max and h_max
- Part (d)(i): Calculate dead load from I-section dimensions (to be assumed or standard ISHB), determine prestressing force P using load balancing concept where parabolic tendon provides upward uniform load w_up = 8Pe/L² balancing total downward load
- Part (d)(ii): Calculate shift of pressure line using the relationship between tendon eccentricity, prestress force, and applied moments, recognizing that pressure line shifts by M/P from tendon line
- Part (e): Calculate design strength of angle section T_d = A_g·f_y/γ_mo, determine design weld strength per mm, use equilibrium of moments about centroid to find L_w1 and L_w2 with throat thickness t_t = K·s = 0.7×6 mm
- For all parts: Maintain consistent units (N, mm, MPa or GPa), apply appropriate IS code provisions (IS 800:2007 for steel, IS 1343 for prestressing), and state assumptions clearly where dimensions are not explicitly given

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies and applies: three-elastic-constant relationship for (a), dual-criteria design philosophy for (b), simultaneous sliding-tipping equilibrium for (c), load balancing and pressure line theory for (d), and eccentric weld group analysis for (e); no conceptual errors in any part	Correct concepts for 3-4 parts with minor errors in one part (e.g., confusing angle of twist units, or incorrect moment arm for tipping); understands basic principles but misapplies in complex scenarios	Major conceptual errors in 2+ parts (e.g., using direct stress formula for torsion, ignoring tipping condition, or treating weld as concentric); fundamental misunderstanding of structural mechanics principles
Numerical accuracy	20%	10	All calculations accurate to 3 significant figures with correct unit conversions (GPa to MPa, degrees to radians); final answers for ν, E, Δd, T_max, F, h, P, shift, L_w1, L_w2 all within acceptable tolerance; proper handling of scientific notation	Correct methodology with minor arithmetic errors in 1-2 parts; acceptable final answers despite intermediate rounding errors; unit conversion errors that don't affect order of magnitude	Significant calculation errors in multiple parts; wrong order of magnitude results; consistent unit confusion (e.g., mixing kN and N without conversion); computational mistakes rendering answers unrealistic
Diagram quality	20%	10	Clear free-body diagram for (c) showing all forces, dimensions, and moment arms with proper labeling; accurate parabolic cable profile with eccentricity marked for (d); weld configuration diagram for (e) showing L_w1, L_w2, and C_xx location; diagrams aid solution directly	Diagrams present but with minor omissions (e.g., missing dimension labels, rough sketches without scale indication); diagrams support answer but require textual clarification; one part may lack required diagram	Missing or seriously flawed diagrams; diagrams that contradict the solution; no attempt to visualize the structural system; poor labeling making diagrams uninterpretable
Step-by-step derivation	20%	10	Each part shows complete derivation: stated formula → substitution with values → intermediate steps → final answer; logical flow from given data to solution; cross-checking where applicable (e.g., verifying E from both stress-strain and G-ν relationship); clear identification of governing criterion in (b)	Derivations present but with skipped steps or 'therefore' statements without justification; some parts show adequate working while others jump to answer; minor logical gaps that don't invalidate the approach	Minimal or no working shown; answers stated without derivation; incorrect formulas used without justification; chaotic presentation making verification impossible; missing critical steps in equilibrium or compatibility equations
Practical interpretation	20%	10	Interprets results meaningfully: comments on whether calculated ν is typical for aluminium (~0.33), checks if shaft design is stiffness or strength governed, discusses safety implications of sliding vs tipping for box stability, explains significance of pressure line shift for prestress efficiency, and verifies weld lengths are practical and meet IS 800 minimums; relates to Indian code contexts	Brief concluding remarks on 2-3 parts; mentions physical significance without elaboration; some awareness of practical constraints but not systematically applied across all parts	No interpretation or discussion of results; purely mathematical exercise without engineering judgment; fails to recognize unrealistic answers (e.g., ν > 0.5, negative dimensions); no code compliance checks

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