Question

A steel rod of diameter 10 mm and 1 m long is heated from 20°C to 120°C, its coefficient of thermal expansion \( \alpha = 12 \times 10^{-6} / \text{K} \) and Young’s Modulus \( E = 200 \, \text{GPa} \). If the rod is free to expand, the thermal stress developed in it is

• A. Zero
• B. 24 MPa
• C. 240 MPa
• D. 2400 MPa

Hint:

Thermal stress is zero when a material is free to expand; stress only develops if expansion is constrained.

Answer 1

The Correct Option is A

Step 1: Understand the concept of thermal stress.
Thermal stress develops in a material when its thermal expansion is constrained. The thermal stress \( \sigma \) is given by: \[ \sigma = E \alpha \Delta T, \] where \( E \) is Young’s Modulus, \( \alpha \) is the coefficient of thermal expansion, and \( \Delta T \) is the temperature change. However, if the rod is free to expand, no stress develops because there is no constraint preventing the expansion. Step 2: Analyze the given condition.
The problem states that the rod is "free to expand." This means there are no external constraints (e.g., fixed ends) to prevent the rod from expanding due to the temperature increase. Therefore, no thermal stress is developed. Step 3: Verify with the formula.
If the rod were constrained (not free to expand), the thermal strain would be: \[ \epsilon = \alpha \Delta T, \] \[ \Delta T = 120 - 20 = 100 \, \text{°C}, \] \[ \epsilon = (12 \times 10^{-6}) \times 100 = 1.2 \times 10^{-3}, \] \[ \sigma = E \epsilon = (200 \times 10^9) \times (1.2 \times 10^{-3}) = 240 \times 10^6 \, \text{Pa} = 240 \, \text{MPa}. \] But since the rod is free to expand, the strain is not resisted, and thus: \[ \sigma = 0. \] Step 4: Select the correct answer.
Since the rod is free to expand, the thermal stress is zero, matching option (1).