Thermal stress in a fully restrained bar: proportionality with temperature change Fill the blank: Thermal stress is ______ proportional to the change in temperature (for no expansion allowed).

Introduction / Context:
Thermal stresses arise when thermal expansion or contraction is restrained. In many practical cases, bars or members cannot freely extend, producing internal stresses that must be considered in design to avoid cracking or yielding.

Given Data / Assumptions:

• Uniform temperature change delta_T.
• Member is fully restrained from changing length.
• Material remains within elastic range with modulus E and coefficient of thermal expansion alpha.

Concept / Approach:
Free thermal strain would be alpha * delta_T. If restraint prevents this strain, an equal and opposite mechanical strain develops to satisfy compatibility. Using Hooke's law, stress equals E times mechanical strain.

Step-by-Step Solution:
Free strain: epsilon_free = alpha * delta_T.Restrained condition enforces epsilon_total = 0.Mechanical strain: epsilon_mech = - alpha * delta_T.Stress: sigma = E * epsilon_mech = - E * alpha * delta_T.Magnitude: |sigma| = E * alpha * delta_T, which is directly proportional to delta_T.

Verification / Alternative check:
If partial restraint exists, stress scales with the effective restraint stiffness; still, for a given restraint, sigma increases linearly with delta_T until yielding or creep reduces stress.

Why Other Options Are Wrong:
Inversely or independent: contradicts the linear relation. Square dependence does not apply in linear elasticity. Saying it depends only on modulus neglects the role of delta_T and alpha.

Common Pitfalls:
Forgetting sign conventions (tension vs compression); ignoring temperature gradients which cause bending rather than pure axial stress.

Final Answer:
directly