an Airbus A320 with a fuselage diameter of 4 meters is on cruising flight. 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 aluminum skin required to limit the fuselage expansion to no more than 4.1 meters, we'll use the relationship between the pressure difference, material properties, and the expansion in a cylindrical pressure vessel.

### Given:
- Fuselage diameter before expansion: \( D_0 = 4 \, \text{m} = 4000 \, \text{mm} \)
- Fuselage diameter after expansion: \( D = 4.1 \, \text{m} = 4100 \, \text{mm} \)
- Change in diameter: \( \Delta D = D - D_0 = 4100 \, \text{mm} - 4000 \, \text{mm} = 100 \, \text{mm} \)
- Elastic modulus: \( E = 20 \, \text{GPa} = 20 \times 10^9 \, \text{Pa} \)
- Poisson's ratio: \( \nu = 0.33 \)
- Cabin pressure at 2400 meters (cruising altitude).
- Assume the external pressure at the cruising altitude is lower than cabin pressure, creating a differential pressure across the fuselage.

### Assumptions:
1. The fuselage behaves like a thin-walled cylindrical pressure vessel.
2. The radial stress is negligible compared to the axial and hoop stresses.

### Step 1: Pressure difference calculation

Assume standard atmospheric pressure at sea level is 101.3 kPa. If we assume that cabin pressure is equivalent to a lower altitude pressure and the external pressure at cruising altitude is lower, the pressure difference is roughly the differential cabin pressure:

\[
\Delta P = P_{\text{cabin}} - P_{\text{outside}}
\]

Without specific pressures provided, we'll assume a typical cabin pressure of approximately 75 kPa at 2400 meters, with an outside pressure lower, let's approximate \( \Delta P = 50 \, \text{kPa} \).

### Step 2: Hoop strain due to internal pressure

The relationship between the hoop stress \(\sigma_h\) in a cylindrical vessel and the internal pressure \( \Delta P \) is given by:

\[
\sigma_h = \frac{\Delta P \cdot R}{t}
\]

Where:
- \( R \) is the radius of the fuselage (\(R = \frac{D_0}{2} = 2000 \, \text{mm}\)).
- \( t \) is the thickness of the skin, which we need to determine.

The hoop strain \(\epsilon_h\) is related to the change in diameter by:

\[
\epsilon_h = \frac{\Delta D}{D_0}
\]

\[
\epsilon_h = \frac{\sigma_h}{E} - \nu \cdot \epsilon_z
\]

Where \( \epsilon_z \) is the axial strain, and for a thin-walled pressure vessel in a symmetric condition, we can neglect the axial expansion. Thus:

\[
\epsilon_h = \frac{\Delta D}{D_0} = \frac{\Delta P \cdot R}{E \cdot t}
\]

Rearranging to solve for \( t \):

\[
t = \frac{\Delta P \cdot R \cdot D_0}{E \cdot \Delta D}
\]

### Step 3: Substitute values

\[
t = \frac{50 \times 10^3 \, \text{Pa} \cdot 2000 \, \text{mm} \cdot 4000 \, \text{mm}}{20 \times 10^9 \, \text{Pa} \cdot 100 \, \text{mm}}
\]

\[
t = \frac{4 \times 10^{11} \, \text{N} \cdot \text{mm}}{2 \times 10^{12} \, \text{N/mm}^2} = 0.2 \, \text{mm}
\]

### Conclusion

The minimum thickness of the aluminum skin should be **0.2 mm**, rounded to the nearest hundreds, to ensure the fuselage does not expand beyond 4.1 meters.

Do you want details or have any questions?

Here are some related questions:
1. How would the required thickness change if the fuselage diameter was increased to 5 meters?
2. What effect does increasing the elastic modulus have on the thickness of the skin?
3. How would a decrease in allowable expansion affect the required skin thickness?
4. What if the Poisson's ratio were different for another material, how would that influence the calculation?
5. How does varying the internal cabin pressure influence the required skin thickness?

**Tip:** For safety and structural integrity in engineering applications, always consider a factor of safety in your final design.

Calculate the minimum thickness of aluminum skin needed for an Airbus A320 fuselage to limit expansion to 4.1 meters during flight. Use mechanics of materials principles, including hoop stress and strain calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation