https://file.aiquickdraw.com/mathbot/file/1730560789018arllw2mt.jpg

To solve this problem, let's analyze the setup in the question.

### Given Data:
1. **Material Properties:**
   - **Aluminum:** 
     - Young's Modulus, \( E_{\text{Al}} = 70 \, \text{GPa} \)
     - Cross-sectional Area, \( A_{\text{Al}} = 900 \, \text{mm}^2 \)
     - Length, \( L_{\text{Al}} = 200 \, \text{mm} \)
   - **Steel:**
     - Young's Modulus, \( E_{\text{Steel}} = 200 \, \text{GPa} \)
     - Cross-sectional Area, \( A_{\text{Steel}} = 1200 \, \text{mm}^2 \)
     - Length, \( L_{\text{Steel}} = 300 \, \text{mm} \)

2. **Objective:** 
   Find the temperature \( T \) at which the aluminum and steel segments have **numerically equal stresses**.

3. **Additional Parameters**:
   - **Thermal expansion coefficients** are not given explicitly in the image, so let's denote them as \( \alpha_{\text{Al}} \) for aluminum and \( \alpha_{\text{Steel}} \) for steel.
   
### Assumptions:
Since the objective is to find the temperature at which the stresses in aluminum and steel are numerically equal, we will assume that:
1. The bar is constrained, meaning the expansion due to temperature increase generates stress in both materials.
2. We will use the relation between thermal strain and thermal stress for both materials.

### Approach:
1. **Thermal Stress** in a constrained bar can be expressed as:
   \[
   \sigma = E \alpha \Delta T
   \]
   where:
   - \( \sigma \) is the thermal stress,
   - \( E \) is the Young's modulus,
   - \( \alpha \) is the coefficient of thermal expansion,
   - \( \Delta T \) is the change in temperature.

2. For **numerically equal stresses** in aluminum and steel:
   \[
   \sigma_{\text{Al}} = \sigma_{\text{Steel}}
   \]
   This translates to:
   \[
   E_{\text{Al}} \alpha_{\text{Al}} \Delta T = E_{\text{Steel}} \alpha_{\text{Steel}} \Delta T
   \]
   
3. Solving for \( \Delta T \), we get:
   \[
   \Delta T = \frac{E_{\text{Steel}} \alpha_{\text{Steel}}}{E_{\text{Al}} \alpha_{\text{Al}}}
   \]

If values for \( \alpha_{\text{Al}} \) and \( \alpha_{\text{Steel}} \) are known, we can substitute them to find the temperature \( \Delta T \) where the stresses are equal.

Would you like to proceed with any assumed values for the thermal expansion coefficients, or do you have additional information about those coefficients? 

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### Follow-up Questions
1. What is the relationship between thermal stress and thermal expansion for composite materials?
2. How does the modulus of elasticity impact the thermal stress in materials of different strengths?
3. Can the length of each segment affect the overall thermal expansion and resulting stresses?
4. How do constraints (like fixed ends) impact the development of thermal stress in materials?
5. How can we apply this principle in real-life scenarios, such as designing multi-material structures?

**Tip:** For multi-material assemblies, always account for different thermal expansion coefficients to avoid stress concentration and possible failure due to temperature changes.

At what temperature will the aluminum and steel segments shown in the figure have numerically equal stresses?

Learn how to determine the temperature at which aluminum and steel segments exhibit equal thermal stresses. This problem involves understanding material properties such as Young's Modulus and thermal expansion coefficients. Suitable for university-level engineering students.

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