To calculate the change in length due to linear expansion, we use the formula:

$\Delta L = L_0 \cdot \alpha \cdot \Delta T$

Where:

• $\Delta L$ = change in length (m)
• $L_0$ = original length (m)
• $\alpha$ = coefficient of linear expansion (per °C)
• $\Delta T$ = change in temperature (°C)

Step 1: Known values

• $L_0 = 9.2 \, \text{m}$
• $\Delta T = 40^\circ \text{C} - 20^\circ \text{C} = 20^\circ \text{C}$
• For steel, \alpha \approx 12 \times 10^{-6} \, \text{per} \, ^\circ \text{C} (approximate value)

Step 2: Calculate the change in length

$\Delta L = 9.2 \cdot (12 \times 10^{-6}) \cdot 20$

$\Delta L = 9.2 \cdot 240 \times 10^{-6}$

$\Delta L = 0.002208 \, \text{m}$

Step 3: Convert to millimeters (if needed)

$\Delta L = 0.002208 \, \text{m} = 2.208 \, \text{mm}$

Final Answer:

The change in length of the steel beam is 2.208 mm.

Would you like additional clarification or calculations?

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Related Questions:

1. How does the coefficient of linear expansion vary among materials?
2. What happens to a material if the temperature drops instead of increases?
3. How would this calculation differ if the beam were aluminum instead of steel?
4. What is the impact of prolonged temperature fluctuations on structural integrity?
5. How can engineers compensate for thermal expansion in bridge design?

Tip:

Always double-check the coefficient of linear expansion ($\alpha$) for the specific material used, as it can vary depending on its composition or alloy type.