Question

Two bars A and B made of different material have same length. The coefficient of expansion and Young's modulus of bar B is twice that of bar A. If the temperature of both bars is increased by the same amount while preventing any expansion, then the ratio of stress developed in bar A to that in bar B is

• A. $16$
• B. $8$
• C. $4$
• D. $2$

Hint:

Thermal stress depends directly on the product of $\alpha$ and $E$ ($\sigma \propto \alpha E$). Since Bar B has twice the value for both properties, it experiences $2 \times 2 = 4$ times more internal thermal stress than Bar A.

Answer 1

The Correct Option is C

Concept: When a bar of length $L$ with a thermal expansion coefficient $\alpha$ undergoes a temperature change $\Delta T$, its free thermal expansion change in length is $\delta_{\text{th}} = \alpha L \Delta T$. If this expansion is completely prevented by rigid boundary supports, a compressive thermal strain develops in the bar: \[ \epsilon = \frac{\delta_{\text{th}}}{L} = \alpha \Delta T \] According to Hooke's Law, the resulting internal thermal stress $\sigma$ depends on the Young's Modulus $E$ of the material: \[ \sigma = E \epsilon = \alpha E \Delta T \]

Step 1: Extracting the proportional relations from the problem statement.
Let us list the material property relationships given for Bar A and Bar B:

• Coefficient of thermal expansion: $\alpha_B = 2\alpha_A$

• Young's Modulus: $E_B = 2E_A$

• Both bars undergo the exact same temperature increase: $\Delta T_A = \Delta T_B = \Delta T$

Step 2: Writing down the stress equations for each bar.
Using the thermal stress formula for Bar A: \[ \sigma_A = \alpha_A E_A \Delta T \] Using the thermal stress formula for Bar B, and substituting its properties in terms of Bar A: \[ \sigma_B = \alpha_B E_B \Delta T = (2\alpha_A)(2E_A)\Delta T = 4 \alpha_A E_A \Delta T \]

Step 3: Finding the ratio between the stresses.
Comparing the two stress equations shows that the stress developed in Bar B is four times larger than the stress in Bar A: \[ \sigma_B = 4\sigma_A \quad \Rightarrow \quad \frac{\sigma_B}{\sigma_A} = 4 \] The ratio between the stresses in the two bars is exactly $4$, which corresponds to Option (C).