GATE ME 2023 Set 1 — Question 46
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Strength of Materials → Stress, Strain & Elastic Constants → Thermal Stresses
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Question
Cylindrical bars P and Q have identical lengths and radii, but are composed of different linear elastic materials. The Young’s modulus and coefficient of thermal expansion of Q are twice the corresponding values of P. Assume the bars to be perfectly bonded at the interface, and their weights to be negligible.The bars are held between rigid supports as shown in the figure and the temperature is raised by . Assume that the stress in each bar is homogeneous and uniaxial. Denote the magnitudes of stress in P and Q by and , respectively.Which of the statement(s) given is/are CORRECT?
Correct answer
(A) The interface between P and Q moves to the left after heating; (D) σ₁ = σ₂
Solution
1.Stress Equality: Since the bars are in series between two rigid supports, the axial force must be the same in both bars. Given they have identical radii (and thus identical cross-sectional areas ), the stress must be equal in both bars. Therefore, . (Option D is correct).
2.Interface Displacement: Let be the length of each bar. Let and be the Young's modulus and thermal expansion coefficient for bar P. Then for bar Q, and .
Total deformation is zero: ..Let be the displacement of the interface to the right. Then :
.
Since is negative, the interface moves to the left. (Option A is correct).
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