Thermal stress in a restrained bar: A circular bar tapers uniformly from diameter d1 at one end to diameter d2 at the other. If its temperature is increased uniformly by t (°C) and the bar is fully restrained against expansion, what is the induced thermal stress? (α = coefficient of linear expansion, E = modulus of elasticity)

Introduction / Context:
Thermal stresses arise when thermal expansion or contraction is restrained. Designers must quantify these stresses to prevent cracking or yielding in components subject to temperature changes, such as turbine shafts or bridge members.

Given Data / Assumptions:

• The bar is circular and tapers linearly from diameter d1 to d2.
• Uniform temperature rise t is applied to the entire bar.
• Both ends are fully restrained so that net axial extension is zero.
• Material is homogeneous and isotropic with modulus E and expansion coefficient α.

Concept / Approach:
If a uniform axial strain α * t would occur freely, but the bar is fully restrained, an equal and opposite mechanical strain must be induced to enforce zero net extension. In uniaxial elasticity, mechanical strain ε_mech = sigma / E. Setting total strain to zero gives sigma / E + α * t = 0, hence sigma = −E * α * t. The magnitude is E * α * t and it is uniform along the bar under full restraint, independent of area variation.

Step-by-Step Solution:

Verification / Alternative check:
A tapered bar has varying stiffness EA(x), but for uniform temperature and full restraint, compatibility requires uniform axial strain (zero total), yielding a uniform stress solution; equilibrium is satisfied without axial load transfer variation.

Why Other Options Are Wrong:

• Area-dependent formulas (options b and d) incorrectly imply geometry dependence for uniform thermal stress under full restraint.
• sigma = 0: only true if the bar is free to expand (unrestrained), which is not the case here.

Common Pitfalls:
Confusing partially restrained conditions (which would require integrating EA(x)) with fully restrained ends; forgetting the sign convention (compressive stress under heating).

Final Answer:
sigma = E * α * t