The problem involves a steel tube and a gunmetal rod enclosed within it. We need to calculate the stresses developed in both materials due to a temperature increase and find the increase in length.

Here's how we approach the problem:

Given dataThe problem involves a steel tube and a gunmetal rod enclosed within it. We need to calculate the stresses developed in both materials due to a temperature increase and find the increase in length.

Here's how we approach the problem:

Given

• Steel tube:
  • Outer diameter: $35 \, \text{mm}$
  • Inner diameter: $30 \, \text{mm}$
• Gunmetal rod:
  • Diameter: $25 \, \text{mm}$
• Initial temperature: $40^\circ C$
• Final temperature: $240^\circ C$
• Coefficient of thermal expansion:
  • Steel: $11 \times 10^{-6}/^\circ C$
  • Gunmetal: $18 \times 10^{-6}/^\circ C$
• Young's modulus:
  • Steel: $205 \, \text{GPa}$
  • Gunmetal: $91.5 \, \text{GPa}$
• Original length: $1 \, \text{m}$

Step 1: Thermal Strain

For each material, the free thermal expansion is given by:

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

Where:

• $L_0 = 1 \, \text{m}$
• $\Delta T = 240^\circ C - 40^\circ C = 200^\circ C$

For steel:

$\Delta L_{\text{steel}} = 1 \times 11 \times 10^{-6} \times 200 = 2.2 \times 10^{-3} \, \text{m}$

For gunmetal:

$\Delta L_{\text{gunmetal}} = 1 \times 18 \times 10^{-6} \times 200 = 3.6 \times 10^{-3} \, \text{m}$

Step 2: Compatibility Condition

Since the assembly is rigidly fixed at the ends, both materials are forced to expand equally. The strain in both materials must be the same, though their free expansions differ. The thermal strains lead to stresses, which will balance each other.

Let $\sigma_s$ be the stress in the steel and $\sigma_g$ be the stress in the gunmetal. The compatibility condition can be expressed as:

$\frac{\sigma_s}{E_s} + \alpha_s \Delta T = \frac{\sigma_g}{E_g} + \alpha_g \Delta T$

Solving this equation allows us to find the stresses in both materials.

Step 3: Stresses in Each Material

Using Young's moduli:

• For steel: $E_s = 205 \, \text{GPa} = 205 \times 10^9 \, \text{Pa}$
• For gunmetal: $E_g = 91.5 \, \text{GPa} = 91.5 \times 10^9 \, \text{Pa}$

We'll use the equations of equilibrium and compatibility to solve for the stresses.

Let me calculate these next.

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Relative questions:

1. How do you calculate thermal stress in constrained systems?
2. Why is thermal expansion important in structural analysis?
3. What factors affect the stress developed due to thermal expansion in materials?
4. How does Young's modulus influence the stresses in different materials?
5. Can this principle of constrained thermal expansion be applied to other systems, such as composites?

Tip: Always account for both thermal expansion and material properties when calculating stress in structures exposed to temperature changes.