Thermal stress in a restrained bar – dependence on area\n\nIn a prismatic bar fully restrained against expansion and subjected to a uniform temperature rise ΔT, the induced thermal stress will

Introduction / Context:
Thermal stresses arise when free thermal expansion or contraction is prevented. Designers must know whether bar size alters the induced stress under full restraint to select materials and allowances properly.

Given Data / Assumptions:

• Uniform temperature change ΔT.
• Bar is perfectly restrained axially (no net change in length).
• Linear elastic material with Young’s modulus E and thermal expansion coefficient α.

Concept / Approach:
Free thermal strain would be ε_free = α ΔT. Full restraint enforces ε_total = 0, so mechanical strain must cancel thermal strain. Hence ε_mech = −α ΔT, and the induced stress is σ = E * ε_mech = −E α ΔT, independent of area.

Step-by-Step Solution:
Free strain: ε_free = α ΔT.Restraint condition: ε_total = ε_mech + ε_free = 0.Therefore ε_mech = −α ΔT.Stress from Hooke’s law: σ = E * ε_mech = −E α ΔT (sign indicates compression for heating).

Verification / Alternative check:
Although the reaction force depends on area (F = σ * A), the stress level σ does not. Two bars of different areas but same material and ΔT develop the same thermal stress under identical perfect restraint.

Why Other Options Are Wrong:
(a) confuses force with stress; (c) moment of inertia matters for bending, not pure axial thermal stress; (d) zero stress occurs only when expansion is unrestrained; (e) length influences free expansion, but not the stress under perfect restraint.

Common Pitfalls:
Mixing up stress and force; assuming partial restraint behaves the same as full restraint; neglecting temperature gradients which can induce bending.

Final Answer:
does not depend on the cross-sectional area of the bar