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We are tasked with determining the inside diameter (\(d\)) of a cast iron tube based on the following information:

- **Axial compressive load**: \( P = 200 \, \text{kN} = 200 \times 10^3 \, \text{N} \)
- **Outside diameter**: \( D = 180 \, \text{mm} = 0.18 \, \text{m} \)
- **Limiting compressive stress**: \( \sigma_{lim} = 40 \, \text{MPa} = 40 \times 10^6 \, \text{Pa} \)

### Step-by-step solution:

1. **Formula for compressive stress**:
   The compressive stress \( \sigma \) on the tube is given by the formula:

   \[
   \sigma = \frac{P}{A}
   \]

   where:
   - \( P \) is the applied load (force),
   - \( A \) is the cross-sectional area of the tube.

2. **Cross-sectional area of the tube**:
   The tube has a circular cross-section, and the area is given by the difference between the outer area and the inner area:

   \[
   A = \frac{\pi}{4} \left( D^2 - d^2 \right)
   \]

   where \( D \) is the outside diameter, and \( d \) is the inside diameter.

3. **Substitute values**:
   The limiting stress is given as \( 40 \, \text{MPa} \), so:

   \[
   \sigma_{lim} = \frac{P}{\frac{\pi}{4} \left( D^2 - d^2 \right)}
   \]

   Plugging in the known values for \( P \) and \( D \):

   \[
   40 \times 10^6 = \frac{200 \times 10^3}{\frac{\pi}{4} \left( 0.18^2 - d^2 \right)}
   \]

4. **Solve for \( d \)**:
   Rearrange the equation to solve for the inside diameter \( d \).

Let me calculate that for you.The inside diameter of the cast iron tube is approximately **161.35 mm**.

Would you like more details on the steps? 

Here are five related questions to deepen your understanding:
1. How would the result change if the compressive load increased?
2. What role does material strength (compressive stress) play in the design of structural components like tubes?
3. How would the calculation differ for a hollow square cross-section instead of a circular one?
4. What methods could be used to test the actual compressive strength of such a tube in practice?
5. How does the wall thickness (difference between inside and outside diameters) affect the structural stability?

**Tip:** Always ensure units are consistent (e.g., converting between millimeters and meters) when applying formulas in engineering problems.

Given the section of a cast iron tube which supports an axial compressive load of 200 kN. Determine its inside diameter if its outside diameter is 180 mm and the limiting compressive stress is 40 MPa.

This problem involves calculating the inside diameter of a cast iron tube subjected to a compressive load of 200 kN. With an outside diameter of 180 mm and a limiting compressive stress of 40 MPa, this solution walks through the step-by-step algebraic and stress analysis to find the tube's internal diameter. Ideal for engineering students.

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