Question

A steel bar is held between two rigid fixed supports. If the temperature of the bar is increased by \(100^\circ\text{C}\), what is the magnitude of the thermal stress induced in the bar? (Coefficient of thermal expansion is \(11\times10^{-6}/^\circ\text{C}\) and Young's modulus of elasticity is \(210\ \text{GPa}\))

• A. \(110\ \text{MPa}\)
• B. \(210\ \text{MPa}\)
• C. \(231\ \text{MPa}\)
• D. \(462\ \text{MPa}\)

Hint:

For a fully restrained bar, \[ \boxed{ \sigma=E\alpha\Delta T. } \] No thermal strain occurs because expansion is completely prevented.

Answer 1

The Correct Option is C

Step 1: Recall the thermal stress formula. When a bar is rigidly fixed at both ends, thermal expansion is completely prevented. The thermal stress developed is \[ \boxed{ \sigma=E\alpha\Delta T, } \] where
• \(E\) = Young's modulus,
• \(\alpha\) = coefficient of thermal expansion,
• \(\Delta T\) = temperature rise.

Step 2: Substitute the given values. Given, \[ E=210\times10^9\ \text{Pa}, \] \[ \alpha=11\times10^{-6}/^\circ\text{C}, \] \[ \Delta T=100^\circ\text{C}. \] Hence, \[ \sigma = 210\times10^9 \times 11\times10^{-6} \times100. \] \[ = 231\times10^6\ \text{Pa}. \] \[ = 231\ \text{MPa}. \] Therefore, \[ \boxed{231\ \text{MPa}} \] is the correct answer. Thus, \[ \boxed{(C)} \] is the correct answer.