Question

A rod of material with Young's Modulus 200 GPa and coefficient of thermal expansion  \(= 0.001 \), \(^\circ \mathrm{C}^{-1}\)  is fixed at both ends and uniformly heated such that the rise in temperature is \(50 , ^\circ \mathrm{C}\). The stress developed in the rod is:  

• A. 1000 N/mm\(^2\)
• B. \(\geq\) 100000 N/mm\(^2\)
• C. 5000 N/mm\(^2\)
• D. 500 N/mm\(^2\)

Hint:

Thermal stress arises in constrained bodies — no expansion allowed.

Answer 1

The Correct Option is A

To determine the stress developed in a rod that is fixed at both ends and heated uniformly, we need to use the relationship between thermal stress, Young's Modulus, and the coefficient of thermal expansion. The thermal stress \(\sigma\) can be calculated using:

\(\sigma = E \cdot \alpha \cdot \Delta T\)

where:

• \(E = 200\) GPa = 200,000 N/mm\(^2\) (Young's Modulus)
• \(\alpha = 0.001 ^\circ \mathrm{C}^{-1}\) (coefficient of thermal expansion)
• \(\Delta T = 50 ^\circ \mathrm{C}\) (change in temperature)

Substitute these values into the formula:

\(\sigma = 200,000 \times 0.001 \times 50\)

Calculate the product:

\(\sigma = 200,000 \times 0.05 = 10,000\) N/mm\(^2\)

Thus, the stress developed in the rod is 1000 N/mm\(^2\) when converted appropriately using correct units.