An Airbus A320 with a fuselage diameter of 4 meters is on cruising ilight. The cabin pressure during flight is set to 2400 meters. If the fuselage is not allowed to expand more than 4.1 meters, determine the minimum thickness of the skin used in the aircraft. Use aluminum with E = 20 GPA and Poisson's ratio of 0.33. (Give the answer in mm and round of the nearest hundreds)

To determine the minimum thickness of the aircraft skin, we will use the formula for circumferential strain in a cylindrical structure due to internal pressure. The problem involves calculating the strain in the fuselage and ensuring that the expansion does not exceed the allowable limit.

### Step 1: Determine the allowable circumferential strain

The circumferential strain \( \epsilon_{\theta} \) can be found using the following relationship:

\[
\epsilon_{\theta} = \frac{\Delta D}{D}
\]

Where:
- \( \Delta D = D_{\text{final}} - D_{\text{initial}} = 4.1 \, \text{m} - 4.0 \, \text{m} = 0.1 \, \text{m} \)
- \( D = D_{\text{initial}} = 4.0 \, \text{m} \)

Thus,

\[
\epsilon_{\theta} = \frac{0.1 \, \text{m}}{4.0 \, \text{m}} = 0.025
\]

### Step 2: Use Hooke's Law for thin-walled cylinders

For a thin-walled cylindrical pressure vessel, the circumferential strain \( \epsilon_{\theta} \) is related to the pressure difference \( \Delta p \), radius \( r \), thickness \( t \), and material properties as follows:

\[
\epsilon_{\theta} = \frac{\Delta p \cdot r}{E \cdot t} - \nu \left( \frac{\Delta p \cdot r}{E \cdot t} \right)
\]

Rearranging for thickness \( t \):

\[
t = \frac{\Delta p \cdot r}{E \cdot \epsilon_{\theta} \cdot (1 - \nu)}
\]

Where:
- \( E = 20 \, \text{GPa} = 20 \times 10^9 \, \text{Pa} \)
- \( \nu = 0.33 \)
- \( r = \frac{D}{2} = \frac{4.0 \, \text{m}}{2} = 2.0 \, \text{m} \)
- \( \epsilon_{\theta} = 0.025 \)

### Step 3: Calculate the pressure difference

The cabin pressure is set to the pressure at 2400 meters, while the outside pressure is the pressure at the cruising altitude. The pressure difference \( \Delta p \) can be estimated using standard atmospheric conditions (assuming sea level pressure is 101325 Pa):

\[
\Delta p \approx p_{\text{cabin}} - p_{\text{outside}}
\]

The outside pressure can be calculated using the barometric formula, but for simplicity, we often use tables or estimations. Assume a rough value:

- Pressure at 2400 meters \( \approx 75,000 \, \text{Pa} \)
- Pressure at cruising altitude \( \approx 20,000 \, \text{Pa} \)

Thus,

\[
\Delta p \approx 75,000 \, \text{Pa} - 20,000 \, \text{Pa} = 55,000 \, \text{Pa}
\]

### Step 4: Solve for the thickness \( t \)

Now substitute all the values into the thickness equation:

\[
t = \frac{55,000 \, \text{Pa} \cdot 2.0 \, \text{m}}{20 \times 10^9 \, \text{Pa} \cdot 0.025 \cdot (1 - 0.33)}
\]

Simplifying:

\[
t = \frac{110,000 \, \text{Pa} \cdot \text{m}}{20 \times 10^9 \, \text{Pa} \cdot 0.025 \cdot 0.67}
\]

\[
t = \frac{110,000}{335 \times 10^6} \, \text{m}
\]

\[
t \approx 0.000328 \, \text{m} = 0.328 \, \text{mm}
\]

Rounding to the nearest hundredth, the thickness \( t \) is **0.33 mm**.

### Final Answer:
The minimum thickness of the aircraft skin should be **0.33 mm**.

Would you like any details or have any questions?

**Relative Questions:**
1. How does Poisson's ratio affect the calculation of skin thickness?
2. What are the effects of varying the cabin pressure on the fuselage's expansion?
3. How would the thickness change if the material used had a different modulus of elasticity?
4. What are the limitations of using the thin-walled cylinder assumption in this context?
5. How do temperature changes during flight affect the required thickness of the fuselage?

**Tip:** Always consider safety factors when designing structural components to account for uncertainties and ensure durability under various operating conditions.

Learn how to calculate the minimum thickness of aircraft skin for an Airbus A320 using mechanics of materials. This problem involves applying Hooke's Law and circumferential strain formula to determine the suitable thickness of aluminum skin, ensuring safety during flight. Suitable for advanced engineering students and professionals.

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