Question

Consider a 250 mm $\times$ 15 mm $\times$ 10 mm steel bar which is free to expand is heated from $15^\circ$C to $40^\circ$C. What will be developed?

• A. Compressive stress
• B. Tensile stress
• C. Shear stress
• D. No stress

Hint:

Thermal stress develops only when expansion or contraction is restrained. Free expansion produces strain but no stress.

Answer 1

The Correct Option is D

Step 1: Understand thermal expansion.
When a material is heated, it tends to expand. Thermal stress develops only if this expansion is restrained.
Step 2: Analyze the given condition.
The steel bar is stated to be free to expand. This means there is no external restraint preventing its thermal expansion.
Step 3: Apply the concept of thermal stress.
Thermal stress is given by:
\[ \sigma = E \alpha \Delta T \]
This formula applies only when expansion is fully or partially restrained.
Since the bar is free, no resistance is offered against expansion.
Step 4: Conclusion.
As there is no restraint, no stress is developed in the bar.

Answer 2

An alternative way to confirm the result is to think about stress purely in terms of internal force equilibrium, rather than only invoking the general formula for thermal stress.

Stress in a bar can only exist if there is an internal resisting force set up within the material. Such a force arises only when something (a support, a rigid boundary, or an adjacent restrained member) prevents the bar from taking up its natural, stress-free expanded length. Since the bar here is explicitly free at both ends, no external body pushes back against its expansion, so no internal resisting force, and therefore no stress, can develop no matter how much the temperature rises.

1. Compressive stress: This would require something squeezing the bar inward as it tries to expand outward, i.e., an external restraint resisting elongation. No such restraint exists for a freely expanding bar, so this cannot occur.
2. Tensile stress: This would require the bar being pulled apart, which is the opposite of what heating (expansion) tends to do, and again needs an external restraint holding the ends fixed. Since the bar is unrestrained, this cannot occur either.
3. Shear stress: Shear arises from relative sliding between adjacent planes of material under transverse or torsional loading; simple free thermal expansion produces uniform elongation along the bar's axis with no such sliding action, so no shear stress is set up.
4. No stress: Since the bar is completely free to expand and nothing resists that expansion, the strain is fully accommodated as a change in dimensions with zero internal resisting force, matching this option exactly.

Because the bar changes size freely without any opposing restraint, the increase in length is pure strain with no accompanying stress.

Therefore, the correct answer is no stress is developed.